state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
F : Type u
K✝¹ : Type v
L : Type w
inst✝³ : Field K✝¹
inst✝² : Field L
inst✝¹ : Field F
n✝ : ℕ
K✝ : Type u
inst✝ : Field K✝
n : ℕ
ih :
(fun n =>
∀ {K : Type u} [inst : Field K] (f : K[X]),
natDegree f = n → Algebra.adjoin K (rootSet f (SplittingFieldAux n f)) = ⊤)
n
K : Type u
x✝ : Field K
f : K[X]
... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | have hmf0 : map (algebraMap K (SplittingFieldAux n.succ f)) f ≠ 0 := map_ne_zero hfn0 | theorem adjoin_rootSet (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤ :=
Nat.recOn (motive := fun n =>
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (... | Mathlib.FieldTheory.SplittingField.Construction.190_0.YlgQXigN9SoLViE | theorem adjoin_rootSet (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤ | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K✝¹ : Type v
L : Type w
inst✝³ : Field K✝¹
inst✝² : Field L
inst✝¹ : Field F
n✝ : ℕ
K✝ : Type u
inst✝ : Field K✝
n : ℕ
ih :
(fun n =>
∀ {K : Type u} [inst : Field K] (f : K[X]),
natDegree f = n → Algebra.adjoin K (rootSet f (SplittingFieldAux n f)) = ⊤)
n
K : Type u
x✝ : Field K
f : K[X]
... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | rw [rootSet_def, aroots_def] | theorem adjoin_rootSet (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤ :=
Nat.recOn (motive := fun n =>
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (... | Mathlib.FieldTheory.SplittingField.Construction.190_0.YlgQXigN9SoLViE | theorem adjoin_rootSet (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤ | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K✝¹ : Type v
L : Type w
inst✝³ : Field K✝¹
inst✝² : Field L
inst✝¹ : Field F
n✝ : ℕ
K✝ : Type u
inst✝ : Field K✝
n : ℕ
ih :
(fun n =>
∀ {K : Type u} [inst : Field K] (f : K[X]),
natDegree f = n → Algebra.adjoin K (rootSet f (SplittingFieldAux n f)) = ⊤)
n
K : Type u
x✝ : Field K
f : K[X]
... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | rw [algebraMap_succ, ← map_map, ← X_sub_C_mul_removeFactor _ hndf, Polynomial.map_mul] at hmf0 ⊢ | theorem adjoin_rootSet (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤ :=
Nat.recOn (motive := fun n =>
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (... | Mathlib.FieldTheory.SplittingField.Construction.190_0.YlgQXigN9SoLViE | theorem adjoin_rootSet (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤ | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K✝¹ : Type v
L : Type w
inst✝³ : Field K✝¹
inst✝² : Field L
inst✝¹ : Field F
n✝ : ℕ
K✝ : Type u
inst✝ : Field K✝
n : ℕ
ih :
(fun n =>
∀ {K : Type u} [inst : Field K] (f : K[X]),
natDegree f = n → Algebra.adjoin K (rootSet f (SplittingFieldAux n f)) = ⊤)
n
K : Type u
x✝ : Field K
f : K[X]
... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | erw [roots_mul hmf0, Polynomial.map_sub, map_X, map_C, roots_X_sub_C, Multiset.toFinset_add,
Finset.coe_union, Multiset.toFinset_singleton, Finset.coe_singleton,
Algebra.adjoin_union_eq_adjoin_adjoin, ← Set.image_singleton,
Algebra.adjoin_algebraMap K (SplittingFieldAux n f.removeFactor),
Adjoin... | theorem adjoin_rootSet (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤ :=
Nat.recOn (motive := fun n =>
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (... | Mathlib.FieldTheory.SplittingField.Construction.190_0.YlgQXigN9SoLViE | theorem adjoin_rootSet (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤ | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K✝¹ : Type v
L : Type w
inst✝³ : Field K✝¹
inst✝² : Field L
inst✝¹ : Field F
n✝ : ℕ
K✝ : Type u
inst✝ : Field K✝
n : ℕ
ih :
(fun n =>
∀ {K : Type u} [inst : Field K] (f : K[X]),
natDegree f = n → Algebra.adjoin K (rootSet f (SplittingFieldAux n f)) = ⊤)
n
K : Type u
x✝ : Field K
f : K[X]
... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | have := IsScalarTower.adjoin_range_toAlgHom K (AdjoinRoot f.factor)
(SplittingFieldAux n f.removeFactor)
(f.removeFactor.rootSet (SplittingFieldAux n f.removeFactor)) | theorem adjoin_rootSet (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤ :=
Nat.recOn (motive := fun n =>
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (... | Mathlib.FieldTheory.SplittingField.Construction.190_0.YlgQXigN9SoLViE | theorem adjoin_rootSet (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤ | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K✝¹ : Type v
L : Type w
inst✝³ : Field K✝¹
inst✝² : Field L
inst✝¹ : Field F
n✝ : ℕ
K✝ : Type u
inst✝ : Field K✝
n : ℕ
ih :
(fun n =>
∀ {K : Type u} [inst : Field K] (f : K[X]),
natDegree f = n → Algebra.adjoin K (rootSet f (SplittingFieldAux n f)) = ⊤)
n
K : Type u
x✝ : Field K
f : K[X]
... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | refine this.trans ?_ | theorem adjoin_rootSet (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤ :=
Nat.recOn (motive := fun n =>
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (... | Mathlib.FieldTheory.SplittingField.Construction.190_0.YlgQXigN9SoLViE | theorem adjoin_rootSet (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤ | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K✝¹ : Type v
L : Type w
inst✝³ : Field K✝¹
inst✝² : Field L
inst✝¹ : Field F
n✝ : ℕ
K✝ : Type u
inst✝ : Field K✝
n : ℕ
ih :
(fun n =>
∀ {K : Type u} [inst : Field K] (f : K[X]),
natDegree f = n → Algebra.adjoin K (rootSet f (SplittingFieldAux n f)) = ⊤)
n
K : Type u
x✝ : Field K
f : K[X]
... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | rw [ih _ (natDegree_removeFactor' hfn), Subalgebra.restrictScalars_top] | theorem adjoin_rootSet (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤ :=
Nat.recOn (motive := fun n =>
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (... | Mathlib.FieldTheory.SplittingField.Construction.190_0.YlgQXigN9SoLViE | theorem adjoin_rootSet (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤ | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f✝ f : K[X]
x : SplittingFieldAux (natDegree f) f
⊢ (MvPolynomial.aeval id) (MvPolynomial.X x) = x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | simp | /-- The algebra equivalence with `SplittingFieldAux`,
which we will use to construct the field structure. -/
def algEquivSplittingFieldAux (f : K[X]) : SplittingField f ≃ₐ[K] SplittingFieldAux f.natDegree f :=
Ideal.quotientKerAlgEquivOfSurjective fun x => ⟨MvPolynomial.X x, by | Mathlib.FieldTheory.SplittingField.Construction.263_0.YlgQXigN9SoLViE | /-- The algebra equivalence with `SplittingFieldAux`,
which we will use to construct the field structure. -/
def algEquivSplittingFieldAux (f : K[X]) : SplittingField f ≃ₐ[K] SplittingFieldAux f.natDegree f | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f : K[X]
e : SplittingField f ≃ₐ[K] SplittingFieldAux (natDegree f) f := algEquivSplittingFieldAux f
a : SplittingField f
w : a ≠ 0
⊢ a * a⁻¹ = 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | apply_fun e | instance : Field (SplittingField f) :=
let e := algEquivSplittingFieldAux f
{ toCommRing := SplittingField.commRing f
ratCast := fun a => algebraMap K _ (a : K)
inv := fun a => e.symm (e a)⁻¹
qsmul := (· • ·)
qsmul_eq_mul' := fun a x =>
Quotient.inductionOn x (fun p => congr_arg Quotient.mk''
... | Mathlib.FieldTheory.SplittingField.Construction.269_0.YlgQXigN9SoLViE | instance : Field (SplittingField f) | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f : K[X]
e : SplittingField f ≃ₐ[K] SplittingFieldAux (natDegree f) f := algEquivSplittingFieldAux f
a : SplittingField f
w : a ≠ 0
⊢ e (a * a⁻¹) = e 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | have : e a ≠ 0 := fun w' => by
apply w
simp? at w' says
simp only [AddEquivClass.map_eq_zero_iff] at w'
exact w' | instance : Field (SplittingField f) :=
let e := algEquivSplittingFieldAux f
{ toCommRing := SplittingField.commRing f
ratCast := fun a => algebraMap K _ (a : K)
inv := fun a => e.symm (e a)⁻¹
qsmul := (· • ·)
qsmul_eq_mul' := fun a x =>
Quotient.inductionOn x (fun p => congr_arg Quotient.mk''
... | Mathlib.FieldTheory.SplittingField.Construction.269_0.YlgQXigN9SoLViE | instance : Field (SplittingField f) | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f : K[X]
e : SplittingField f ≃ₐ[K] SplittingFieldAux (natDegree f) f := algEquivSplittingFieldAux f
a : SplittingField f
w : a ≠ 0
w' : e a = 0
⊢ False | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | apply w | instance : Field (SplittingField f) :=
let e := algEquivSplittingFieldAux f
{ toCommRing := SplittingField.commRing f
ratCast := fun a => algebraMap K _ (a : K)
inv := fun a => e.symm (e a)⁻¹
qsmul := (· • ·)
qsmul_eq_mul' := fun a x =>
Quotient.inductionOn x (fun p => congr_arg Quotient.mk''
... | Mathlib.FieldTheory.SplittingField.Construction.269_0.YlgQXigN9SoLViE | instance : Field (SplittingField f) | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f : K[X]
e : SplittingField f ≃ₐ[K] SplittingFieldAux (natDegree f) f := algEquivSplittingFieldAux f
a : SplittingField f
w : a ≠ 0
w' : e a = 0
⊢ a = 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | simp? at w' says
simp only [AddEquivClass.map_eq_zero_iff] at w' | instance : Field (SplittingField f) :=
let e := algEquivSplittingFieldAux f
{ toCommRing := SplittingField.commRing f
ratCast := fun a => algebraMap K _ (a : K)
inv := fun a => e.symm (e a)⁻¹
qsmul := (· • ·)
qsmul_eq_mul' := fun a x =>
Quotient.inductionOn x (fun p => congr_arg Quotient.mk''
... | Mathlib.FieldTheory.SplittingField.Construction.269_0.YlgQXigN9SoLViE | instance : Field (SplittingField f) | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f : K[X]
e : SplittingField f ≃ₐ[K] SplittingFieldAux (natDegree f) f := algEquivSplittingFieldAux f
a : SplittingField f
w : a ≠ 0
w' : e a = 0
⊢ a = 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | simp only [AddEquivClass.map_eq_zero_iff] at w' | instance : Field (SplittingField f) :=
let e := algEquivSplittingFieldAux f
{ toCommRing := SplittingField.commRing f
ratCast := fun a => algebraMap K _ (a : K)
inv := fun a => e.symm (e a)⁻¹
qsmul := (· • ·)
qsmul_eq_mul' := fun a x =>
Quotient.inductionOn x (fun p => congr_arg Quotient.mk''
... | Mathlib.FieldTheory.SplittingField.Construction.269_0.YlgQXigN9SoLViE | instance : Field (SplittingField f) | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f : K[X]
e : SplittingField f ≃ₐ[K] SplittingFieldAux (natDegree f) f := algEquivSplittingFieldAux f
a : SplittingField f
w : a ≠ 0
w' : a = 0
⊢ a = 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | exact w' | instance : Field (SplittingField f) :=
let e := algEquivSplittingFieldAux f
{ toCommRing := SplittingField.commRing f
ratCast := fun a => algebraMap K _ (a : K)
inv := fun a => e.symm (e a)⁻¹
qsmul := (· • ·)
qsmul_eq_mul' := fun a x =>
Quotient.inductionOn x (fun p => congr_arg Quotient.mk''
... | Mathlib.FieldTheory.SplittingField.Construction.269_0.YlgQXigN9SoLViE | instance : Field (SplittingField f) | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f : K[X]
e : SplittingField f ≃ₐ[K] SplittingFieldAux (natDegree f) f := algEquivSplittingFieldAux f
a : SplittingField f
w : a ≠ 0
this : e a ≠ 0
⊢ e (a * a⁻¹) = e 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | simp only [map_mul, AlgEquiv.apply_symm_apply, ne_eq, AddEquivClass.map_eq_zero_iff, map_one] | instance : Field (SplittingField f) :=
let e := algEquivSplittingFieldAux f
{ toCommRing := SplittingField.commRing f
ratCast := fun a => algebraMap K _ (a : K)
inv := fun a => e.symm (e a)⁻¹
qsmul := (· • ·)
qsmul_eq_mul' := fun a x =>
Quotient.inductionOn x (fun p => congr_arg Quotient.mk''
... | Mathlib.FieldTheory.SplittingField.Construction.269_0.YlgQXigN9SoLViE | instance : Field (SplittingField f) | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f : K[X]
e : SplittingField f ≃ₐ[K] SplittingFieldAux (natDegree f) f := algEquivSplittingFieldAux f
a : SplittingField f
w : a ≠ 0
this : e a ≠ 0
⊢ (algEquivSplittingFieldAux f) a * ((algEquivSplittingFieldAux f) a)⁻¹ = 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | rw [mul_inv_cancel] | instance : Field (SplittingField f) :=
let e := algEquivSplittingFieldAux f
{ toCommRing := SplittingField.commRing f
ratCast := fun a => algebraMap K _ (a : K)
inv := fun a => e.symm (e a)⁻¹
qsmul := (· • ·)
qsmul_eq_mul' := fun a x =>
Quotient.inductionOn x (fun p => congr_arg Quotient.mk''
... | Mathlib.FieldTheory.SplittingField.Construction.269_0.YlgQXigN9SoLViE | instance : Field (SplittingField f) | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f : K[X]
e : SplittingField f ≃ₐ[K] SplittingFieldAux (natDegree f) f := algEquivSplittingFieldAux f
a : SplittingField f
w : a ≠ 0
this : e a ≠ 0
⊢ (algEquivSplittingFieldAux f) a ≠ 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | assumption | instance : Field (SplittingField f) :=
let e := algEquivSplittingFieldAux f
{ toCommRing := SplittingField.commRing f
ratCast := fun a => algebraMap K _ (a : K)
inv := fun a => e.symm (e a)⁻¹
qsmul := (· • ·)
qsmul_eq_mul' := fun a x =>
Quotient.inductionOn x (fun p => congr_arg Quotient.mk''
... | Mathlib.FieldTheory.SplittingField.Construction.269_0.YlgQXigN9SoLViE | instance : Field (SplittingField f) | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f : K[X]
e : SplittingField f ≃ₐ[K] SplittingFieldAux (natDegree f) f := algEquivSplittingFieldAux f
⊢ 0⁻¹ = 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | simp | instance : Field (SplittingField f) :=
let e := algEquivSplittingFieldAux f
{ toCommRing := SplittingField.commRing f
ratCast := fun a => algebraMap K _ (a : K)
inv := fun a => e.symm (e a)⁻¹
qsmul := (· • ·)
qsmul_eq_mul' := fun a x =>
Quotient.inductionOn x (fun p => congr_arg Quotient.mk''
... | Mathlib.FieldTheory.SplittingField.Construction.269_0.YlgQXigN9SoLViE | instance : Field (SplittingField f) | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f : K[X]
e : SplittingField f ≃ₐ[K] SplittingFieldAux (natDegree f) f := algEquivSplittingFieldAux f
a : ℤ
b : ℕ
h1 : b ≠ 0
h2 : Nat.Coprime (Int.natAbs a) b
⊢ ↑(Rat.mk' a b) = ↑a * (↑b)⁻¹ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | apply_fun e | instance : Field (SplittingField f) :=
let e := algEquivSplittingFieldAux f
{ toCommRing := SplittingField.commRing f
ratCast := fun a => algebraMap K _ (a : K)
inv := fun a => e.symm (e a)⁻¹
qsmul := (· • ·)
qsmul_eq_mul' := fun a x =>
Quotient.inductionOn x (fun p => congr_arg Quotient.mk''
... | Mathlib.FieldTheory.SplittingField.Construction.269_0.YlgQXigN9SoLViE | instance : Field (SplittingField f) | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f : K[X]
e : SplittingField f ≃ₐ[K] SplittingFieldAux (natDegree f) f := algEquivSplittingFieldAux f
a : ℤ
b : ℕ
h1 : b ≠ 0
h2 : Nat.Coprime (Int.natAbs a) b
⊢ e ↑(Rat.mk' a b) = e (↑a * (↑b)⁻¹) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | change e (algebraMap K _ _) = _ | instance : Field (SplittingField f) :=
let e := algEquivSplittingFieldAux f
{ toCommRing := SplittingField.commRing f
ratCast := fun a => algebraMap K _ (a : K)
inv := fun a => e.symm (e a)⁻¹
qsmul := (· • ·)
qsmul_eq_mul' := fun a x =>
Quotient.inductionOn x (fun p => congr_arg Quotient.mk''
... | Mathlib.FieldTheory.SplittingField.Construction.269_0.YlgQXigN9SoLViE | instance : Field (SplittingField f) | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f : K[X]
e : SplittingField f ≃ₐ[K] SplittingFieldAux (natDegree f) f := algEquivSplittingFieldAux f
a : ℤ
b : ℕ
h1 : b ≠ 0
h2 : Nat.Coprime (Int.natAbs a) b
⊢ e ((algebraMap K (SplittingField f)) ↑(Rat.mk' a b)) = e (↑a * (↑b)⁻¹) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | simp only [map_ratCast, map_natCast, map_mul, map_intCast, AlgEquiv.commutes,
AlgEquiv.apply_symm_apply] | instance : Field (SplittingField f) :=
let e := algEquivSplittingFieldAux f
{ toCommRing := SplittingField.commRing f
ratCast := fun a => algebraMap K _ (a : K)
inv := fun a => e.symm (e a)⁻¹
qsmul := (· • ·)
qsmul_eq_mul' := fun a x =>
Quotient.inductionOn x (fun p => congr_arg Quotient.mk''
... | Mathlib.FieldTheory.SplittingField.Construction.269_0.YlgQXigN9SoLViE | instance : Field (SplittingField f) | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f : K[X]
e : SplittingField f ≃ₐ[K] SplittingFieldAux (natDegree f) f := algEquivSplittingFieldAux f
a : ℤ
b : ℕ
h1 : b ≠ 0
h2 : Nat.Coprime (Int.natAbs a) b
⊢ ↑(Rat.mk' a b) = ↑a * (↑b)⁻¹ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | apply Field.ratCast_mk | instance : Field (SplittingField f) :=
let e := algEquivSplittingFieldAux f
{ toCommRing := SplittingField.commRing f
ratCast := fun a => algebraMap K _ (a : K)
inv := fun a => e.symm (e a)⁻¹
qsmul := (· • ·)
qsmul_eq_mul' := fun a x =>
Quotient.inductionOn x (fun p => congr_arg Quotient.mk''
... | Mathlib.FieldTheory.SplittingField.Construction.269_0.YlgQXigN9SoLViE | instance : Field (SplittingField f) | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f : K[X]
e : SplittingField f ≃ₐ[K] SplittingFieldAux (natDegree f) f := algEquivSplittingFieldAux f
a : ℚ
x : SplittingField f
p : MvPolynomial (SplittingFieldAux (natDegree f) f) K
⊢ (fun x => a • x) p = (fun x x_1 => x * x_1) ((algebr... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | ext | instance : Field (SplittingField f) :=
let e := algEquivSplittingFieldAux f
{ toCommRing := SplittingField.commRing f
ratCast := fun a => algebraMap K _ (a : K)
inv := fun a => e.symm (e a)⁻¹
qsmul := (· • ·)
qsmul_eq_mul' := fun a x =>
Quotient.inductionOn x (fun p => congr_arg Quotient.mk''
... | Mathlib.FieldTheory.SplittingField.Construction.269_0.YlgQXigN9SoLViE | instance : Field (SplittingField f) | Mathlib_FieldTheory_SplittingField_Construction |
case a
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f : K[X]
e : SplittingField f ≃ₐ[K] SplittingFieldAux (natDegree f) f := algEquivSplittingFieldAux f
a : ℚ
x : SplittingField f
p : MvPolynomial (SplittingFieldAux (natDegree f) f) K
m✝ : SplittingFieldAux (natDegree f) f →₀ ℕ
⊢ M... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | simp [MvPolynomial.algebraMap_eq, Rat.smul_def] | instance : Field (SplittingField f) :=
let e := algEquivSplittingFieldAux f
{ toCommRing := SplittingField.commRing f
ratCast := fun a => algebraMap K _ (a : K)
inv := fun a => e.symm (e a)⁻¹
qsmul := (· • ·)
qsmul_eq_mul' := fun a x =>
Quotient.inductionOn x (fun p => congr_arg Quotient.mk''
... | Mathlib.FieldTheory.SplittingField.Construction.269_0.YlgQXigN9SoLViE | instance : Field (SplittingField f) | Mathlib_FieldTheory_SplittingField_Construction |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : SigmaFinite ν
f :... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | simp_rw [Integrable, hf.of_uncurry_left.aestronglyMeasurable, true_and_iff] | theorem measurableSet_integrable [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : MeasurableSet {x | Integrable (f x) ν} := by
| Mathlib.MeasureTheory.Constructions.Prod.Integral.67_0.4tLli7hg1Kzhb8p | theorem measurableSet_integrable [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : MeasurableSet {x | Integrable (f x) ν} | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : SigmaFinite ν
f :... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | exact measurableSet_lt (Measurable.lintegral_prod_right hf.ennnorm) measurable_const | theorem measurableSet_integrable [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : MeasurableSet {x | Integrable (f x) ν} := by
simp_rw [Integrable, hf.of_uncurry_left.aestronglyMeasurable, true_and_iff]
| Mathlib.MeasureTheory.Constructions.Prod.Integral.67_0.4tLli7hg1Kzhb8p | theorem measurableSet_integrable [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : MeasurableSet {x | Integrable (f x) ν} | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | by_cases hE : CompleteSpace E | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case pos
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedS... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | swap | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case neg
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedS... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | simp [integral, hE, stronglyMeasurable_const] | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case pos
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedS... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | borelize E | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case pos
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedS... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | haveI : SeparableSpace (range (uncurry f) ∪ {0} : Set E) :=
hf.separableSpace_range_union_singleton | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case pos
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedS... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | let s : ℕ → SimpleFunc (α × β) E :=
SimpleFunc.approxOn _ hf.measurable (range (uncurry f) ∪ {0}) 0 (by simp) | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | simp | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case pos
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedS... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | let s' : ℕ → α → SimpleFunc β E := fun n x => (s n).comp (Prod.mk x) measurable_prod_mk_left | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case pos
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedS... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | let f' : ℕ → α → E := fun n => {x | Integrable (f x) ν}.indicator fun x => (s' n x).integral ν | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case pos
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedS... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | have hf' : ∀ n, StronglyMeasurable (f' n) := by
intro n; refine' StronglyMeasurable.indicator _ (measurableSet_integrable hf)
have : ∀ x, ((s' n x).range.filter fun x => x ≠ 0) ⊆ (s n).range := by
intro x; refine' Finset.Subset.trans (Finset.filter_subset _ _) _; intro y
simp_rw [SimpleFunc.mem_rang... | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | intro n | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | refine' StronglyMeasurable.indicator _ (measurableSet_integrable hf) | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | have : ∀ x, ((s' n x).range.filter fun x => x ≠ 0) ⊆ (s n).range := by
intro x; refine' Finset.Subset.trans (Finset.filter_subset _ _) _; intro y
simp_rw [SimpleFunc.mem_range]; rintro ⟨z, rfl⟩; exact ⟨(x, z), rfl⟩ | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | intro x | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | refine' Finset.Subset.trans (Finset.filter_subset _ _) _ | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | intro y | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | simp_rw [SimpleFunc.mem_range] | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | rintro ⟨z, rfl⟩ | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case intro
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : Norme... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | exact ⟨(x, z), rfl⟩ | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | simp only [SimpleFunc.integral_eq_sum_of_subset (this _)] | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | refine' Finset.stronglyMeasurable_sum _ fun x _ => _ | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | refine' (Measurable.ennreal_toReal _).stronglyMeasurable.smul_const _ | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | simp (config := { singlePass := true }) only [SimpleFunc.coe_comp, preimage_comp] | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | apply measurable_measure_prod_mk_left | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case hs
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSp... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | exact (s n).measurableSet_fiber x | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case pos
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedS... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | have h2f' : Tendsto f' atTop (𝓝 fun x : α => ∫ y : β, f x y ∂ν) := by
rw [tendsto_pi_nhds]; intro x
by_cases hfx : Integrable (f x) ν
· have : ∀ n, Integrable (s' n x) ν := by
intro n; apply (hfx.norm.add hfx.norm).mono' (s' n x).aestronglyMeasurable
apply eventually_of_forall; intro y
... | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | rw [tendsto_pi_nhds] | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | intro x | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | by_cases hfx : Integrable (f x) ν | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case pos
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedS... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | have : ∀ n, Integrable (s' n x) ν := by
intro n; apply (hfx.norm.add hfx.norm).mono' (s' n x).aestronglyMeasurable
apply eventually_of_forall; intro y
simp_rw [SimpleFunc.coe_comp]; exact SimpleFunc.norm_approxOn_zero_le _ _ (x, y) n | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | intro n | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | apply (hfx.norm.add hfx.norm).mono' (s' n x).aestronglyMeasurable | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | apply eventually_of_forall | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case hp
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSp... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | intro y | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case hp
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSp... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | simp_rw [SimpleFunc.coe_comp] | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case hp
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSp... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | exact SimpleFunc.norm_approxOn_zero_le _ _ (x, y) n | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case pos
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedS... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | simp only [hfx, SimpleFunc.integral_eq_integral _ (this _), indicator_of_mem,
mem_setOf_eq] | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case pos
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedS... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | refine'
tendsto_integral_of_dominated_convergence (fun y => ‖f x y‖ + ‖f x y‖)
(fun n => (s' n x).aestronglyMeasurable) (hfx.norm.add hfx.norm) _ _ | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case pos.refine'_1
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | refine' fun n => eventually_of_forall fun y => SimpleFunc.norm_approxOn_zero_le _ _ (x, y) n | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case pos.refine'_1.refine'_1
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGrou... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | exact hf.measurable | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case pos.refine'_1.refine'_2
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGrou... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | simp | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case pos.refine'_2
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | refine' eventually_of_forall fun y => SimpleFunc.tendsto_approxOn _ _ _ | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case pos.refine'_2.refine'_1
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGrou... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | exact hf.measurable.of_uncurry_left | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case pos.refine'_2.refine'_2
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGrou... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | simp | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case pos.refine'_2.refine'_3
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGrou... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | apply subset_closure | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case pos.refine'_2.refine'_3.a
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGr... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | simp [-uncurry_apply_pair] | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case neg
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedS... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | simp [hfx, integral_undef] | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case pos
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedS... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | exact stronglyMeasurable_of_tendsto _ hf' h2f' | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.77_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : Strongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | rw [← uncurry_curry f] at hf | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right' [SigmaFinite ν] ⦃f : α × β → E⦄
(hf : StronglyMeasurable f) : StronglyMeasurable fun x => ∫ y, f (x, y) ∂ν := by
| Mathlib.MeasureTheory.Constructions.Prod.Integral.127_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right' [SigmaFinite ν] ⦃f : α × β → E⦄
(hf : StronglyMeasurable f) : StronglyMeasurable fun x => ∫ y, f (x, y) ∂ν | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | exact hf.integral_prod_right | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right' [SigmaFinite ν] ⦃f : α × β → E⦄
(hf : StronglyMeasurable f) : StronglyMeasurable fun x => ∫ y, f (x, y) ∂ν := by
rw [... | Mathlib.MeasureTheory.Constructions.Prod.Integral.127_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right' [SigmaFinite ν] ⦃f : α × β → E⦄
(hf : StronglyMeasurable f) : StronglyMeasurable fun x => ∫ y, f (x, y) ∂ν | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : SigmaFinite ν
s :... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | refine' ⟨(measurable_measure_prod_mk_left hs).ennreal_toReal.aemeasurable.aestronglyMeasurable, _⟩ | theorem integrable_measure_prod_mk_left {s : Set (α × β)} (hs : MeasurableSet s)
(h2s : (μ.prod ν) s ≠ ∞) : Integrable (fun x => (ν (Prod.mk x ⁻¹' s)).toReal) μ := by
| Mathlib.MeasureTheory.Constructions.Prod.Integral.160_0.4tLli7hg1Kzhb8p | theorem integrable_measure_prod_mk_left {s : Set (α × β)} (hs : MeasurableSet s)
(h2s : (μ.prod ν) s ≠ ∞) : Integrable (fun x => (ν (Prod.mk x ⁻¹' s)).toReal) μ | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : SigmaFinite ν
s :... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | simp_rw [HasFiniteIntegral, ennnorm_eq_ofReal toReal_nonneg] | theorem integrable_measure_prod_mk_left {s : Set (α × β)} (hs : MeasurableSet s)
(h2s : (μ.prod ν) s ≠ ∞) : Integrable (fun x => (ν (Prod.mk x ⁻¹' s)).toReal) μ := by
refine' ⟨(measurable_measure_prod_mk_left hs).ennreal_toReal.aemeasurable.aestronglyMeasurable, _⟩
| Mathlib.MeasureTheory.Constructions.Prod.Integral.160_0.4tLli7hg1Kzhb8p | theorem integrable_measure_prod_mk_left {s : Set (α × β)} (hs : MeasurableSet s)
(h2s : (μ.prod ν) s ≠ ∞) : Integrable (fun x => (ν (Prod.mk x ⁻¹' s)).toReal) μ | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : SigmaFinite ν
s :... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | convert h2s.lt_top using 1 | theorem integrable_measure_prod_mk_left {s : Set (α × β)} (hs : MeasurableSet s)
(h2s : (μ.prod ν) s ≠ ∞) : Integrable (fun x => (ν (Prod.mk x ⁻¹' s)).toReal) μ := by
refine' ⟨(measurable_measure_prod_mk_left hs).ennreal_toReal.aemeasurable.aestronglyMeasurable, _⟩
simp_rw [HasFiniteIntegral, ennnorm_eq_ofReal ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.160_0.4tLli7hg1Kzhb8p | theorem integrable_measure_prod_mk_left {s : Set (α × β)} (hs : MeasurableSet s)
(h2s : (μ.prod ν) s ≠ ∞) : Integrable (fun x => (ν (Prod.mk x ⁻¹' s)).toReal) μ | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case h.e'_3
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : Sigma... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | rw [prod_apply hs] | theorem integrable_measure_prod_mk_left {s : Set (α × β)} (hs : MeasurableSet s)
(h2s : (μ.prod ν) s ≠ ∞) : Integrable (fun x => (ν (Prod.mk x ⁻¹' s)).toReal) μ := by
refine' ⟨(measurable_measure_prod_mk_left hs).ennreal_toReal.aemeasurable.aestronglyMeasurable, _⟩
simp_rw [HasFiniteIntegral, ennnorm_eq_ofReal ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.160_0.4tLli7hg1Kzhb8p | theorem integrable_measure_prod_mk_left {s : Set (α × β)} (hs : MeasurableSet s)
(h2s : (μ.prod ν) s ≠ ∞) : Integrable (fun x => (ν (Prod.mk x ⁻¹' s)).toReal) μ | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case h.e'_3
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : Sigma... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | apply lintegral_congr_ae | theorem integrable_measure_prod_mk_left {s : Set (α × β)} (hs : MeasurableSet s)
(h2s : (μ.prod ν) s ≠ ∞) : Integrable (fun x => (ν (Prod.mk x ⁻¹' s)).toReal) μ := by
refine' ⟨(measurable_measure_prod_mk_left hs).ennreal_toReal.aemeasurable.aestronglyMeasurable, _⟩
simp_rw [HasFiniteIntegral, ennnorm_eq_ofReal ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.160_0.4tLli7hg1Kzhb8p | theorem integrable_measure_prod_mk_left {s : Set (α × β)} (hs : MeasurableSet s)
(h2s : (μ.prod ν) s ≠ ∞) : Integrable (fun x => (ν (Prod.mk x ⁻¹' s)).toReal) μ | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case h.e'_3.h
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : Sig... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | refine' (ae_measure_lt_top hs h2s).mp _ | theorem integrable_measure_prod_mk_left {s : Set (α × β)} (hs : MeasurableSet s)
(h2s : (μ.prod ν) s ≠ ∞) : Integrable (fun x => (ν (Prod.mk x ⁻¹' s)).toReal) μ := by
refine' ⟨(measurable_measure_prod_mk_left hs).ennreal_toReal.aemeasurable.aestronglyMeasurable, _⟩
simp_rw [HasFiniteIntegral, ennnorm_eq_ofReal ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.160_0.4tLli7hg1Kzhb8p | theorem integrable_measure_prod_mk_left {s : Set (α × β)} (hs : MeasurableSet s)
(h2s : (μ.prod ν) s ≠ ∞) : Integrable (fun x => (ν (Prod.mk x ⁻¹' s)).toReal) μ | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case h.e'_3.h
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : Sig... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | apply eventually_of_forall | theorem integrable_measure_prod_mk_left {s : Set (α × β)} (hs : MeasurableSet s)
(h2s : (μ.prod ν) s ≠ ∞) : Integrable (fun x => (ν (Prod.mk x ⁻¹' s)).toReal) μ := by
refine' ⟨(measurable_measure_prod_mk_left hs).ennreal_toReal.aemeasurable.aestronglyMeasurable, _⟩
simp_rw [HasFiniteIntegral, ennnorm_eq_ofReal ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.160_0.4tLli7hg1Kzhb8p | theorem integrable_measure_prod_mk_left {s : Set (α × β)} (hs : MeasurableSet s)
(h2s : (μ.prod ν) s ≠ ∞) : Integrable (fun x => (ν (Prod.mk x ⁻¹' s)).toReal) μ | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case h.e'_3.h.hp
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : ... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | intro x hx | theorem integrable_measure_prod_mk_left {s : Set (α × β)} (hs : MeasurableSet s)
(h2s : (μ.prod ν) s ≠ ∞) : Integrable (fun x => (ν (Prod.mk x ⁻¹' s)).toReal) μ := by
refine' ⟨(measurable_measure_prod_mk_left hs).ennreal_toReal.aemeasurable.aestronglyMeasurable, _⟩
simp_rw [HasFiniteIntegral, ennnorm_eq_ofReal ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.160_0.4tLli7hg1Kzhb8p | theorem integrable_measure_prod_mk_left {s : Set (α × β)} (hs : MeasurableSet s)
(h2s : (μ.prod ν) s ≠ ∞) : Integrable (fun x => (ν (Prod.mk x ⁻¹' s)).toReal) μ | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case h.e'_3.h.hp
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : ... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | rw [lt_top_iff_ne_top] at hx | theorem integrable_measure_prod_mk_left {s : Set (α × β)} (hs : MeasurableSet s)
(h2s : (μ.prod ν) s ≠ ∞) : Integrable (fun x => (ν (Prod.mk x ⁻¹' s)).toReal) μ := by
refine' ⟨(measurable_measure_prod_mk_left hs).ennreal_toReal.aemeasurable.aestronglyMeasurable, _⟩
simp_rw [HasFiniteIntegral, ennnorm_eq_ofReal ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.160_0.4tLli7hg1Kzhb8p | theorem integrable_measure_prod_mk_left {s : Set (α × β)} (hs : MeasurableSet s)
(h2s : (μ.prod ν) s ≠ ∞) : Integrable (fun x => (ν (Prod.mk x ⁻¹' s)).toReal) μ | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case h.e'_3.h.hp
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : ... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | simp [ofReal_toReal, hx] | theorem integrable_measure_prod_mk_left {s : Set (α × β)} (hs : MeasurableSet s)
(h2s : (μ.prod ν) s ≠ ∞) : Integrable (fun x => (ν (Prod.mk x ⁻¹' s)).toReal) μ := by
refine' ⟨(measurable_measure_prod_mk_left hs).ennreal_toReal.aemeasurable.aestronglyMeasurable, _⟩
simp_rw [HasFiniteIntegral, ennnorm_eq_ofReal ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.160_0.4tLli7hg1Kzhb8p | theorem integrable_measure_prod_mk_left {s : Set (α × β)} (hs : MeasurableSet s)
(h2s : (μ.prod ν) s ≠ ∞) : Integrable (fun x => (ν (Prod.mk x ⁻¹' s)).toReal) μ | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ✝ : Type u_5
E : Type u_6
inst✝⁸ : MeasurableSpace α
inst✝⁷ : MeasurableSpace α'
inst✝⁶ : MeasurableSpace β
inst✝⁵ : MeasurableSpace β'
inst✝⁴ : MeasurableSpace γ✝
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ✝
inst✝³ : NormedAddCommGroup E
γ : Type u_7
inst✝² : ... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | rw [← prod_swap] at hf | nonrec theorem MeasureTheory.AEStronglyMeasurable.prod_swap {γ : Type*} [TopologicalSpace γ]
[SigmaFinite μ] [SigmaFinite ν] {f : β × α → γ} (hf : AEStronglyMeasurable f (ν.prod μ)) :
AEStronglyMeasurable (fun z : α × β => f z.swap) (μ.prod ν) := by
| Mathlib.MeasureTheory.Constructions.Prod.Integral.182_0.4tLli7hg1Kzhb8p | nonrec theorem MeasureTheory.AEStronglyMeasurable.prod_swap {γ : Type*} [TopologicalSpace γ]
[SigmaFinite μ] [SigmaFinite ν] {f : β × α → γ} (hf : AEStronglyMeasurable f (ν.prod μ)) :
AEStronglyMeasurable (fun z : α × β => f z.swap) (μ.prod ν) | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ✝ : Type u_5
E : Type u_6
inst✝⁸ : MeasurableSpace α
inst✝⁷ : MeasurableSpace α'
inst✝⁶ : MeasurableSpace β
inst✝⁵ : MeasurableSpace β'
inst✝⁴ : MeasurableSpace γ✝
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ✝
inst✝³ : NormedAddCommGroup E
γ : Type u_7
inst✝² : ... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | exact hf.comp_measurable measurable_swap | nonrec theorem MeasureTheory.AEStronglyMeasurable.prod_swap {γ : Type*} [TopologicalSpace γ]
[SigmaFinite μ] [SigmaFinite ν] {f : β × α → γ} (hf : AEStronglyMeasurable f (ν.prod μ)) :
AEStronglyMeasurable (fun z : α × β => f z.swap) (μ.prod ν) := by
rw [← prod_swap] at hf
| Mathlib.MeasureTheory.Constructions.Prod.Integral.182_0.4tLli7hg1Kzhb8p | nonrec theorem MeasureTheory.AEStronglyMeasurable.prod_swap {γ : Type*} [TopologicalSpace γ]
[SigmaFinite μ] [SigmaFinite ν] {f : β × α → γ} (hf : AEStronglyMeasurable f (ν.prod μ)) :
AEStronglyMeasurable (fun z : α × β => f z.swap) (μ.prod ν) | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : SigmaFinite ν
in... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | filter_upwards [ae_ae_of_ae_prod hf.ae_eq_mk] with _ hx using integral_congr_ae hx | /-- The Bochner integral is a.e.-measurable.
This shows that the integrand of (the right-hand-side of) Fubini's theorem is a.e.-measurable. -/
theorem MeasureTheory.AEStronglyMeasurable.integral_prod_right' [SigmaFinite ν] [NormedSpace ℝ E]
⦃f : α × β → E⦄ (hf : AEStronglyMeasurable f (μ.prod ν)) :
AEStrongly... | Mathlib.MeasureTheory.Constructions.Prod.Integral.199_0.4tLli7hg1Kzhb8p | /-- The Bochner integral is a.e.-measurable.
This shows that the integrand of (the right-hand-side of) Fubini's theorem is a.e.-measurable. -/
theorem MeasureTheory.AEStronglyMeasurable.integral_prod_right' [SigmaFinite ν] [NormedSpace ℝ E]
⦃f : α × β → E⦄ (hf : AEStronglyMeasurable f (μ.prod ν)) :
AEStrongly... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ✝ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ✝
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ✝
inst✝² : NormedAddCommGroup E
γ : Type u_7
inst✝¹ : ... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | filter_upwards [ae_ae_of_ae_prod hf.ae_eq_mk] with x hx | theorem MeasureTheory.AEStronglyMeasurable.prod_mk_left {γ : Type*} [SigmaFinite ν]
[TopologicalSpace γ] {f : α × β → γ} (hf : AEStronglyMeasurable f (μ.prod ν)) :
∀ᵐ x ∂μ, AEStronglyMeasurable (fun y => f (x, y)) ν := by
| Mathlib.MeasureTheory.Constructions.Prod.Integral.208_0.4tLli7hg1Kzhb8p | theorem MeasureTheory.AEStronglyMeasurable.prod_mk_left {γ : Type*} [SigmaFinite ν]
[TopologicalSpace γ] {f : α × β → γ} (hf : AEStronglyMeasurable f (μ.prod ν)) :
∀ᵐ x ∂μ, AEStronglyMeasurable (fun y => f (x, y)) ν | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case h
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ✝ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ✝
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ✝
inst✝² : NormedAddCommGroup E
γ : Type u_7
in... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | exact
⟨fun y => hf.mk f (x, y), hf.stronglyMeasurable_mk.comp_measurable measurable_prod_mk_left, hx⟩ | theorem MeasureTheory.AEStronglyMeasurable.prod_mk_left {γ : Type*} [SigmaFinite ν]
[TopologicalSpace γ] {f : α × β → γ} (hf : AEStronglyMeasurable f (μ.prod ν)) :
∀ᵐ x ∂μ, AEStronglyMeasurable (fun y => f (x, y)) ν := by
filter_upwards [ae_ae_of_ae_prod hf.ae_eq_mk] with x hx
| Mathlib.MeasureTheory.Constructions.Prod.Integral.208_0.4tLli7hg1Kzhb8p | theorem MeasureTheory.AEStronglyMeasurable.prod_mk_left {γ : Type*} [SigmaFinite ν]
[TopologicalSpace γ] {f : α × β → γ} (hf : AEStronglyMeasurable f (μ.prod ν)) :
∀ᵐ x ∂μ, AEStronglyMeasurable (fun y => f (x, y)) ν | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : SigmaFinite ν
f :... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | simp only [HasFiniteIntegral] | theorem hasFiniteIntegral_prod_iff ⦃f : α × β → E⦄ (h1f : StronglyMeasurable f) :
HasFiniteIntegral f (μ.prod ν) ↔
(∀ᵐ x ∂μ, HasFiniteIntegral (fun y => f (x, y)) ν) ∧
HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂ν) μ := by
| Mathlib.MeasureTheory.Constructions.Prod.Integral.236_0.4tLli7hg1Kzhb8p | theorem hasFiniteIntegral_prod_iff ⦃f : α × β → E⦄ (h1f : StronglyMeasurable f) :
HasFiniteIntegral f (μ.prod ν) ↔
(∀ᵐ x ∂μ, HasFiniteIntegral (fun y => f (x, y)) ν) ∧
HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂ν) μ | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : SigmaFinite ν
f :... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | rw [lintegral_prod_of_measurable _ h1f.ennnorm] | theorem hasFiniteIntegral_prod_iff ⦃f : α × β → E⦄ (h1f : StronglyMeasurable f) :
HasFiniteIntegral f (μ.prod ν) ↔
(∀ᵐ x ∂μ, HasFiniteIntegral (fun y => f (x, y)) ν) ∧
HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂ν) μ := by
simp only [HasFiniteIntegral]
-- Porting note: was `simp`
| Mathlib.MeasureTheory.Constructions.Prod.Integral.236_0.4tLli7hg1Kzhb8p | theorem hasFiniteIntegral_prod_iff ⦃f : α × β → E⦄ (h1f : StronglyMeasurable f) :
HasFiniteIntegral f (μ.prod ν) ↔
(∀ᵐ x ∂μ, HasFiniteIntegral (fun y => f (x, y)) ν) ∧
HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂ν) μ | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : SigmaFinite ν
f :... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | have : ∀ x, ∀ᵐ y ∂ν, 0 ≤ ‖f (x, y)‖ := fun x => eventually_of_forall fun y => norm_nonneg _ | theorem hasFiniteIntegral_prod_iff ⦃f : α × β → E⦄ (h1f : StronglyMeasurable f) :
HasFiniteIntegral f (μ.prod ν) ↔
(∀ᵐ x ∂μ, HasFiniteIntegral (fun y => f (x, y)) ν) ∧
HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂ν) μ := by
simp only [HasFiniteIntegral]
-- Porting note: was `simp`
rw [lintegral_... | Mathlib.MeasureTheory.Constructions.Prod.Integral.236_0.4tLli7hg1Kzhb8p | theorem hasFiniteIntegral_prod_iff ⦃f : α × β → E⦄ (h1f : StronglyMeasurable f) :
HasFiniteIntegral f (μ.prod ν) ↔
(∀ᵐ x ∂μ, HasFiniteIntegral (fun y => f (x, y)) ν) ∧
HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂ν) μ | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : SigmaFinite ν
f :... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | simp_rw [integral_eq_lintegral_of_nonneg_ae (this _)
(h1f.norm.comp_measurable measurable_prod_mk_left).aestronglyMeasurable,
ennnorm_eq_ofReal toReal_nonneg, ofReal_norm_eq_coe_nnnorm] | theorem hasFiniteIntegral_prod_iff ⦃f : α × β → E⦄ (h1f : StronglyMeasurable f) :
HasFiniteIntegral f (μ.prod ν) ↔
(∀ᵐ x ∂μ, HasFiniteIntegral (fun y => f (x, y)) ν) ∧
HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂ν) μ := by
simp only [HasFiniteIntegral]
-- Porting note: was `simp`
rw [lintegral_... | Mathlib.MeasureTheory.Constructions.Prod.Integral.236_0.4tLli7hg1Kzhb8p | theorem hasFiniteIntegral_prod_iff ⦃f : α × β → E⦄ (h1f : StronglyMeasurable f) :
HasFiniteIntegral f (μ.prod ν) ↔
(∀ᵐ x ∂μ, HasFiniteIntegral (fun y => f (x, y)) ν) ∧
HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂ν) μ | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : SigmaFinite ν
f :... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | have : ∀ {p q r : Prop} (_ : r → p), (r ↔ p ∧ q) ↔ p → (r ↔ q) := fun {p q r} h1 => by
rw [← and_congr_right_iff, and_iff_right_of_imp h1] | theorem hasFiniteIntegral_prod_iff ⦃f : α × β → E⦄ (h1f : StronglyMeasurable f) :
HasFiniteIntegral f (μ.prod ν) ↔
(∀ᵐ x ∂μ, HasFiniteIntegral (fun y => f (x, y)) ν) ∧
HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂ν) μ := by
simp only [HasFiniteIntegral]
-- Porting note: was `simp`
rw [lintegral_... | Mathlib.MeasureTheory.Constructions.Prod.Integral.236_0.4tLli7hg1Kzhb8p | theorem hasFiniteIntegral_prod_iff ⦃f : α × β → E⦄ (h1f : StronglyMeasurable f) :
HasFiniteIntegral f (μ.prod ν) ↔
(∀ᵐ x ∂μ, HasFiniteIntegral (fun y => f (x, y)) ν) ∧
HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂ν) μ | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : SigmaFinite ν
f :... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | rw [← and_congr_right_iff, and_iff_right_of_imp h1] | theorem hasFiniteIntegral_prod_iff ⦃f : α × β → E⦄ (h1f : StronglyMeasurable f) :
HasFiniteIntegral f (μ.prod ν) ↔
(∀ᵐ x ∂μ, HasFiniteIntegral (fun y => f (x, y)) ν) ∧
HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂ν) μ := by
simp only [HasFiniteIntegral]
-- Porting note: was `simp`
rw [lintegral_... | Mathlib.MeasureTheory.Constructions.Prod.Integral.236_0.4tLli7hg1Kzhb8p | theorem hasFiniteIntegral_prod_iff ⦃f : α × β → E⦄ (h1f : StronglyMeasurable f) :
HasFiniteIntegral f (μ.prod ν) ↔
(∀ᵐ x ∂μ, HasFiniteIntegral (fun y => f (x, y)) ν) ∧
HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂ν) μ | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : SigmaFinite ν
f :... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | rw [this] | theorem hasFiniteIntegral_prod_iff ⦃f : α × β → E⦄ (h1f : StronglyMeasurable f) :
HasFiniteIntegral f (μ.prod ν) ↔
(∀ᵐ x ∂μ, HasFiniteIntegral (fun y => f (x, y)) ν) ∧
HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂ν) μ := by
simp only [HasFiniteIntegral]
-- Porting note: was `simp`
rw [lintegral_... | Mathlib.MeasureTheory.Constructions.Prod.Integral.236_0.4tLli7hg1Kzhb8p | theorem hasFiniteIntegral_prod_iff ⦃f : α × β → E⦄ (h1f : StronglyMeasurable f) :
HasFiniteIntegral f (μ.prod ν) ↔
(∀ᵐ x ∂μ, HasFiniteIntegral (fun y => f (x, y)) ν) ∧
HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂ν) μ | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : SigmaFinite ν
f :... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | intro h2f | theorem hasFiniteIntegral_prod_iff ⦃f : α × β → E⦄ (h1f : StronglyMeasurable f) :
HasFiniteIntegral f (μ.prod ν) ↔
(∀ᵐ x ∂μ, HasFiniteIntegral (fun y => f (x, y)) ν) ∧
HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂ν) μ := by
simp only [HasFiniteIntegral]
-- Porting note: was `simp`
rw [lintegral_... | Mathlib.MeasureTheory.Constructions.Prod.Integral.236_0.4tLli7hg1Kzhb8p | theorem hasFiniteIntegral_prod_iff ⦃f : α × β → E⦄ (h1f : StronglyMeasurable f) :
HasFiniteIntegral f (μ.prod ν) ↔
(∀ᵐ x ∂μ, HasFiniteIntegral (fun y => f (x, y)) ν) ∧
HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂ν) μ | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : SigmaFinite ν
f :... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | rw [lintegral_congr_ae] | theorem hasFiniteIntegral_prod_iff ⦃f : α × β → E⦄ (h1f : StronglyMeasurable f) :
HasFiniteIntegral f (μ.prod ν) ↔
(∀ᵐ x ∂μ, HasFiniteIntegral (fun y => f (x, y)) ν) ∧
HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂ν) μ := by
simp only [HasFiniteIntegral]
-- Porting note: was `simp`
rw [lintegral_... | Mathlib.MeasureTheory.Constructions.Prod.Integral.236_0.4tLli7hg1Kzhb8p | theorem hasFiniteIntegral_prod_iff ⦃f : α × β → E⦄ (h1f : StronglyMeasurable f) :
HasFiniteIntegral f (μ.prod ν) ↔
(∀ᵐ x ∂μ, HasFiniteIntegral (fun y => f (x, y)) ν) ∧
HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂ν) μ | Mathlib_MeasureTheory_Constructions_Prod_Integral |
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