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 |
|---|---|---|---|---|---|---|
case this.zero
⊢ ∀ (s : ℂ), -s.re < ↑Nat.zero → DifferentiableAt ℂ (fun u => (Gamma u)⁻¹) s | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | intro s hs | /-- The reciprocal of the Gamma function is differentiable everywhere (including the points where
Gamma itself is not). -/
theorem differentiable_one_div_Gamma : Differentiable ℂ fun s : ℂ => (Gamma s)⁻¹ := by
suffices : ∀ n : ℕ, ∀ (s : ℂ) (_ : -s.re < n), DifferentiableAt ℂ (fun u : ℂ => (Gamma u)⁻¹) s
exact fun s... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.535_0.in2QiCFW52coQT2 | /-- The reciprocal of the Gamma function is differentiable everywhere (including the points where
Gamma itself is not). -/
theorem differentiable_one_div_Gamma : Differentiable ℂ fun s : ℂ => (Gamma s)⁻¹ | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case this.zero
s : ℂ
hs : -s.re < ↑Nat.zero
⊢ DifferentiableAt ℂ (fun u => (Gamma u)⁻¹) s | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [Nat.cast_zero, neg_lt_zero] at hs | /-- The reciprocal of the Gamma function is differentiable everywhere (including the points where
Gamma itself is not). -/
theorem differentiable_one_div_Gamma : Differentiable ℂ fun s : ℂ => (Gamma s)⁻¹ := by
suffices : ∀ n : ℕ, ∀ (s : ℂ) (_ : -s.re < n), DifferentiableAt ℂ (fun u : ℂ => (Gamma u)⁻¹) s
exact fun s... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.535_0.in2QiCFW52coQT2 | /-- The reciprocal of the Gamma function is differentiable everywhere (including the points where
Gamma itself is not). -/
theorem differentiable_one_div_Gamma : Differentiable ℂ fun s : ℂ => (Gamma s)⁻¹ | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case this.zero
s : ℂ
hs : 0 < s.re
⊢ DifferentiableAt ℂ (fun u => (Gamma u)⁻¹) s | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | suffices : ∀ m : ℕ, s ≠ -↑m | /-- The reciprocal of the Gamma function is differentiable everywhere (including the points where
Gamma itself is not). -/
theorem differentiable_one_div_Gamma : Differentiable ℂ fun s : ℂ => (Gamma s)⁻¹ := by
suffices : ∀ n : ℕ, ∀ (s : ℂ) (_ : -s.re < n), DifferentiableAt ℂ (fun u : ℂ => (Gamma u)⁻¹) s
exact fun s... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.535_0.in2QiCFW52coQT2 | /-- The reciprocal of the Gamma function is differentiable everywhere (including the points where
Gamma itself is not). -/
theorem differentiable_one_div_Gamma : Differentiable ℂ fun s : ℂ => (Gamma s)⁻¹ | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case this.zero
s : ℂ
hs : 0 < s.re
this : ∀ (m : ℕ), s ≠ -↑m
⊢ DifferentiableAt ℂ (fun u => (Gamma u)⁻¹) s
case this s : ℂ hs : 0 < s.re ⊢ ∀ (m : ℕ), s ≠ -↑m | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | exact (differentiableAt_Gamma _ this).inv (Gamma_ne_zero this) | /-- The reciprocal of the Gamma function is differentiable everywhere (including the points where
Gamma itself is not). -/
theorem differentiable_one_div_Gamma : Differentiable ℂ fun s : ℂ => (Gamma s)⁻¹ := by
suffices : ∀ n : ℕ, ∀ (s : ℂ) (_ : -s.re < n), DifferentiableAt ℂ (fun u : ℂ => (Gamma u)⁻¹) s
exact fun s... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.535_0.in2QiCFW52coQT2 | /-- The reciprocal of the Gamma function is differentiable everywhere (including the points where
Gamma itself is not). -/
theorem differentiable_one_div_Gamma : Differentiable ℂ fun s : ℂ => (Gamma s)⁻¹ | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case this
s : ℂ
hs : 0 < s.re
⊢ ∀ (m : ℕ), s ≠ -↑m | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | contrapose! hs | /-- The reciprocal of the Gamma function is differentiable everywhere (including the points where
Gamma itself is not). -/
theorem differentiable_one_div_Gamma : Differentiable ℂ fun s : ℂ => (Gamma s)⁻¹ := by
suffices : ∀ n : ℕ, ∀ (s : ℂ) (_ : -s.re < n), DifferentiableAt ℂ (fun u : ℂ => (Gamma u)⁻¹) s
exact fun s... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.535_0.in2QiCFW52coQT2 | /-- The reciprocal of the Gamma function is differentiable everywhere (including the points where
Gamma itself is not). -/
theorem differentiable_one_div_Gamma : Differentiable ℂ fun s : ℂ => (Gamma s)⁻¹ | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case this
s : ℂ
hs : ∃ m, s = -↑m
⊢ s.re ≤ 0 | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rcases hs with ⟨m, rfl⟩ | /-- The reciprocal of the Gamma function is differentiable everywhere (including the points where
Gamma itself is not). -/
theorem differentiable_one_div_Gamma : Differentiable ℂ fun s : ℂ => (Gamma s)⁻¹ := by
suffices : ∀ n : ℕ, ∀ (s : ℂ) (_ : -s.re < n), DifferentiableAt ℂ (fun u : ℂ => (Gamma u)⁻¹) s
exact fun s... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.535_0.in2QiCFW52coQT2 | /-- The reciprocal of the Gamma function is differentiable everywhere (including the points where
Gamma itself is not). -/
theorem differentiable_one_div_Gamma : Differentiable ℂ fun s : ℂ => (Gamma s)⁻¹ | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case this.intro
m : ℕ
⊢ (-↑m).re ≤ 0 | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | simpa only [neg_re, ← ofReal_nat_cast, ofReal_re, neg_nonpos] using Nat.cast_nonneg m | /-- The reciprocal of the Gamma function is differentiable everywhere (including the points where
Gamma itself is not). -/
theorem differentiable_one_div_Gamma : Differentiable ℂ fun s : ℂ => (Gamma s)⁻¹ := by
suffices : ∀ n : ℕ, ∀ (s : ℂ) (_ : -s.re < n), DifferentiableAt ℂ (fun u : ℂ => (Gamma u)⁻¹) s
exact fun s... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.535_0.in2QiCFW52coQT2 | /-- The reciprocal of the Gamma function is differentiable everywhere (including the points where
Gamma itself is not). -/
theorem differentiable_one_div_Gamma : Differentiable ℂ fun s : ℂ => (Gamma s)⁻¹ | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case this.succ
m : ℕ
hm : ∀ (s : ℂ), -s.re < ↑m → DifferentiableAt ℂ (fun u => (Gamma u)⁻¹) s
⊢ ∀ (s : ℂ), -s.re < ↑(Nat.succ m) → DifferentiableAt ℂ (fun u => (Gamma u)⁻¹) s | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | intro s hs | /-- The reciprocal of the Gamma function is differentiable everywhere (including the points where
Gamma itself is not). -/
theorem differentiable_one_div_Gamma : Differentiable ℂ fun s : ℂ => (Gamma s)⁻¹ := by
suffices : ∀ n : ℕ, ∀ (s : ℂ) (_ : -s.re < n), DifferentiableAt ℂ (fun u : ℂ => (Gamma u)⁻¹) s
exact fun s... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.535_0.in2QiCFW52coQT2 | /-- The reciprocal of the Gamma function is differentiable everywhere (including the points where
Gamma itself is not). -/
theorem differentiable_one_div_Gamma : Differentiable ℂ fun s : ℂ => (Gamma s)⁻¹ | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case this.succ
m : ℕ
hm : ∀ (s : ℂ), -s.re < ↑m → DifferentiableAt ℂ (fun u => (Gamma u)⁻¹) s
s : ℂ
hs : -s.re < ↑(Nat.succ m)
⊢ DifferentiableAt ℂ (fun u => (Gamma u)⁻¹) s | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [funext one_div_Gamma_eq_self_mul_one_div_Gamma_add_one] | /-- The reciprocal of the Gamma function is differentiable everywhere (including the points where
Gamma itself is not). -/
theorem differentiable_one_div_Gamma : Differentiable ℂ fun s : ℂ => (Gamma s)⁻¹ := by
suffices : ∀ n : ℕ, ∀ (s : ℂ) (_ : -s.re < n), DifferentiableAt ℂ (fun u : ℂ => (Gamma u)⁻¹) s
exact fun s... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.535_0.in2QiCFW52coQT2 | /-- The reciprocal of the Gamma function is differentiable everywhere (including the points where
Gamma itself is not). -/
theorem differentiable_one_div_Gamma : Differentiable ℂ fun s : ℂ => (Gamma s)⁻¹ | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case this.succ
m : ℕ
hm : ∀ (s : ℂ), -s.re < ↑m → DifferentiableAt ℂ (fun u => (Gamma u)⁻¹) s
s : ℂ
hs : -s.re < ↑(Nat.succ m)
⊢ DifferentiableAt ℂ (fun x => x * (Gamma (x + 1))⁻¹) s | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | specialize hm (s + 1) (by rwa [add_re, one_re, neg_add', sub_lt_iff_lt_add, ← Nat.cast_succ]) | /-- The reciprocal of the Gamma function is differentiable everywhere (including the points where
Gamma itself is not). -/
theorem differentiable_one_div_Gamma : Differentiable ℂ fun s : ℂ => (Gamma s)⁻¹ := by
suffices : ∀ n : ℕ, ∀ (s : ℂ) (_ : -s.re < n), DifferentiableAt ℂ (fun u : ℂ => (Gamma u)⁻¹) s
exact fun s... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.535_0.in2QiCFW52coQT2 | /-- The reciprocal of the Gamma function is differentiable everywhere (including the points where
Gamma itself is not). -/
theorem differentiable_one_div_Gamma : Differentiable ℂ fun s : ℂ => (Gamma s)⁻¹ | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
m : ℕ
hm : ∀ (s : ℂ), -s.re < ↑m → DifferentiableAt ℂ (fun u => (Gamma u)⁻¹) s
s : ℂ
hs : -s.re < ↑(Nat.succ m)
⊢ -(s + 1).re < ↑m | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rwa [add_re, one_re, neg_add', sub_lt_iff_lt_add, ← Nat.cast_succ] | /-- The reciprocal of the Gamma function is differentiable everywhere (including the points where
Gamma itself is not). -/
theorem differentiable_one_div_Gamma : Differentiable ℂ fun s : ℂ => (Gamma s)⁻¹ := by
suffices : ∀ n : ℕ, ∀ (s : ℂ) (_ : -s.re < n), DifferentiableAt ℂ (fun u : ℂ => (Gamma u)⁻¹) s
exact fun s... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.535_0.in2QiCFW52coQT2 | /-- The reciprocal of the Gamma function is differentiable everywhere (including the points where
Gamma itself is not). -/
theorem differentiable_one_div_Gamma : Differentiable ℂ fun s : ℂ => (Gamma s)⁻¹ | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case this.succ
m : ℕ
s : ℂ
hs : -s.re < ↑(Nat.succ m)
hm : DifferentiableAt ℂ (fun u => (Gamma u)⁻¹) (s + 1)
⊢ DifferentiableAt ℂ (fun x => x * (Gamma (x + 1))⁻¹) s | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | refine' differentiableAt_id.mul (hm.comp s _) | /-- The reciprocal of the Gamma function is differentiable everywhere (including the points where
Gamma itself is not). -/
theorem differentiable_one_div_Gamma : Differentiable ℂ fun s : ℂ => (Gamma s)⁻¹ := by
suffices : ∀ n : ℕ, ∀ (s : ℂ) (_ : -s.re < n), DifferentiableAt ℂ (fun u : ℂ => (Gamma u)⁻¹) s
exact fun s... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.535_0.in2QiCFW52coQT2 | /-- The reciprocal of the Gamma function is differentiable everywhere (including the points where
Gamma itself is not). -/
theorem differentiable_one_div_Gamma : Differentiable ℂ fun s : ℂ => (Gamma s)⁻¹ | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case this.succ
m : ℕ
s : ℂ
hs : -s.re < ↑(Nat.succ m)
hm : DifferentiableAt ℂ (fun u => (Gamma u)⁻¹) (s + 1)
⊢ DifferentiableAt ℂ (fun x => x + 1) s | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | exact differentiableAt_id.add (differentiableAt_const _) | /-- The reciprocal of the Gamma function is differentiable everywhere (including the points where
Gamma itself is not). -/
theorem differentiable_one_div_Gamma : Differentiable ℂ fun s : ℂ => (Gamma s)⁻¹ := by
suffices : ∀ n : ℕ, ∀ (s : ℂ) (_ : -s.re < n), DifferentiableAt ℂ (fun u : ℂ => (Gamma u)⁻¹) s
exact fun s... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.535_0.in2QiCFW52coQT2 | /-- The reciprocal of the Gamma function is differentiable everywhere (including the points where
Gamma itself is not). -/
theorem differentiable_one_div_Gamma : Differentiable ℂ fun s : ℂ => (Gamma s)⁻¹ | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
⊢ Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * ↑(sqrt π) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | suffices (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z =>
(Gamma (2 * z))⁻¹ * (2 : ℂ) ^ (2 * z - 1) / ↑(Real.sqrt π) by
convert congr_arg Inv.inv (congr_fun this s) using 1
· rw [mul_inv, inv_inv, inv_inv]
· rw [div_eq_mul_inv, mul_inv, mul_inv, inv_inv, inv_inv, ← cpow_neg, neg_sub] | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) := by
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.574_0.in2QiCFW52coQT2 | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
this : (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z => (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑(sqrt π)
⊢ Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * ↑(sqrt π) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | convert congr_arg Inv.inv (congr_fun this s) using 1 | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) := by
suffices (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z =>
(Gamma (2 * z))⁻¹ * (2 : ℂ) ^ (2 * z - 1) / ↑(Real.sqrt π) by
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.574_0.in2QiCFW52coQT2 | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case h.e'_2
s : ℂ
this : (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z => (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑(sqrt π)
⊢ Gamma s * Gamma (s + 1 / 2) = ((Gamma s)⁻¹ * (Gamma (s + 1 / 2))⁻¹)⁻¹ | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [mul_inv, inv_inv, inv_inv] | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) := by
suffices (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z =>
(Gamma (2 * z))⁻¹ * (2 : ℂ) ^ (2 * z - 1) / ↑(Real.sqrt π) by
convert congr_arg Inv.inv (congr_fun this... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.574_0.in2QiCFW52coQT2 | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case h.e'_3
s : ℂ
this : (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z => (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑(sqrt π)
⊢ Gamma (2 * s) * 2 ^ (1 - 2 * s) * ↑(sqrt π) = ((Gamma (2 * s))⁻¹ * 2 ^ (2 * s - 1) / ↑(sqrt π))⁻¹ | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [div_eq_mul_inv, mul_inv, mul_inv, inv_inv, inv_inv, ← cpow_neg, neg_sub] | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) := by
suffices (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z =>
(Gamma (2 * z))⁻¹ * (2 : ℂ) ^ (2 * z - 1) / ↑(Real.sqrt π) by
convert congr_arg Inv.inv (congr_fun this... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.574_0.in2QiCFW52coQT2 | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
⊢ (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z => (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑(sqrt π) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | have h1 : AnalyticOn ℂ (fun z : ℂ => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ := by
refine' DifferentiableOn.analyticOn _ isOpen_univ
refine' (differentiable_one_div_Gamma.mul _).differentiableOn
exact differentiable_one_div_Gamma.comp (differentiable_id.add (differentiable_const _)) | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) := by
suffices (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z =>
(Gamma (2 * z))⁻¹ * (2 : ℂ) ^ (2 * z - 1) / ↑(Real.sqrt π) by
convert congr_arg Inv.inv (congr_fun this... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.574_0.in2QiCFW52coQT2 | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
⊢ AnalyticOn ℂ (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | refine' DifferentiableOn.analyticOn _ isOpen_univ | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) := by
suffices (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z =>
(Gamma (2 * z))⁻¹ * (2 : ℂ) ^ (2 * z - 1) / ↑(Real.sqrt π) by
convert congr_arg Inv.inv (congr_fun this... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.574_0.in2QiCFW52coQT2 | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
⊢ DifferentiableOn ℂ (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | refine' (differentiable_one_div_Gamma.mul _).differentiableOn | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) := by
suffices (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z =>
(Gamma (2 * z))⁻¹ * (2 : ℂ) ^ (2 * z - 1) / ↑(Real.sqrt π) by
convert congr_arg Inv.inv (congr_fun this... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.574_0.in2QiCFW52coQT2 | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
⊢ Differentiable ℂ fun z => (Gamma (z + 1 / 2))⁻¹ | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | exact differentiable_one_div_Gamma.comp (differentiable_id.add (differentiable_const _)) | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) := by
suffices (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z =>
(Gamma (2 * z))⁻¹ * (2 : ℂ) ^ (2 * z - 1) / ↑(Real.sqrt π) by
convert congr_arg Inv.inv (congr_fun this... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.574_0.in2QiCFW52coQT2 | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
h1 : AnalyticOn ℂ (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ
⊢ (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z => (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑(sqrt π) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | have h2 : AnalyticOn ℂ
(fun z => (Gamma (2 * z))⁻¹ * (2 : ℂ) ^ (2 * z - 1) / ↑(Real.sqrt π)) univ := by
refine' DifferentiableOn.analyticOn _ isOpen_univ
refine' (Differentiable.mul _ (differentiable_const _)).differentiableOn
apply Differentiable.mul
· exact differentiable_one_div_Gamma.comp (dif... | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) := by
suffices (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z =>
(Gamma (2 * z))⁻¹ * (2 : ℂ) ^ (2 * z - 1) / ↑(Real.sqrt π) by
convert congr_arg Inv.inv (congr_fun this... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.574_0.in2QiCFW52coQT2 | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
h1 : AnalyticOn ℂ (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ
⊢ AnalyticOn ℂ (fun z => (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑(sqrt π)) univ | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | refine' DifferentiableOn.analyticOn _ isOpen_univ | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) := by
suffices (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z =>
(Gamma (2 * z))⁻¹ * (2 : ℂ) ^ (2 * z - 1) / ↑(Real.sqrt π) by
convert congr_arg Inv.inv (congr_fun this... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.574_0.in2QiCFW52coQT2 | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
h1 : AnalyticOn ℂ (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ
⊢ DifferentiableOn ℂ (fun z => (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑(sqrt π)) univ | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | refine' (Differentiable.mul _ (differentiable_const _)).differentiableOn | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) := by
suffices (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z =>
(Gamma (2 * z))⁻¹ * (2 : ℂ) ^ (2 * z - 1) / ↑(Real.sqrt π) by
convert congr_arg Inv.inv (congr_fun this... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.574_0.in2QiCFW52coQT2 | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
h1 : AnalyticOn ℂ (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ
⊢ Differentiable ℂ fun z => (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | apply Differentiable.mul | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) := by
suffices (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z =>
(Gamma (2 * z))⁻¹ * (2 : ℂ) ^ (2 * z - 1) / ↑(Real.sqrt π) by
convert congr_arg Inv.inv (congr_fun this... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.574_0.in2QiCFW52coQT2 | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case ha
s : ℂ
h1 : AnalyticOn ℂ (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ
⊢ Differentiable ℂ fun y => (Gamma (2 * y))⁻¹ | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | exact differentiable_one_div_Gamma.comp (differentiable_id'.const_mul _) | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) := by
suffices (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z =>
(Gamma (2 * z))⁻¹ * (2 : ℂ) ^ (2 * z - 1) / ↑(Real.sqrt π) by
convert congr_arg Inv.inv (congr_fun this... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.574_0.in2QiCFW52coQT2 | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hb
s : ℂ
h1 : AnalyticOn ℂ (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ
⊢ Differentiable ℂ fun y => 2 ^ (2 * y - 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | refine' fun t => DifferentiableAt.const_cpow _ (Or.inl two_ne_zero) | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) := by
suffices (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z =>
(Gamma (2 * z))⁻¹ * (2 : ℂ) ^ (2 * z - 1) / ↑(Real.sqrt π) by
convert congr_arg Inv.inv (congr_fun this... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.574_0.in2QiCFW52coQT2 | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hb
s : ℂ
h1 : AnalyticOn ℂ (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ
t : ℂ
⊢ DifferentiableAt ℂ (fun y => 2 * y - 1) t | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | refine' DifferentiableAt.sub_const (differentiableAt_id.const_mul _) _ | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) := by
suffices (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z =>
(Gamma (2 * z))⁻¹ * (2 : ℂ) ^ (2 * z - 1) / ↑(Real.sqrt π) by
convert congr_arg Inv.inv (congr_fun this... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.574_0.in2QiCFW52coQT2 | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
h1 : AnalyticOn ℂ (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ
h2 : AnalyticOn ℂ (fun z => (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑(sqrt π)) univ
⊢ (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z => (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑(sqrt π) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | have h3 : Tendsto ((↑) : ℝ → ℂ) (𝓝[≠] 1) (𝓝[≠] 1) := by
rw [tendsto_nhdsWithin_iff]; constructor
· exact tendsto_nhdsWithin_of_tendsto_nhds continuous_ofReal.continuousAt
· exact eventually_nhdsWithin_iff.mpr (eventually_of_forall fun t ht => ofReal_ne_one.mpr ht) | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) := by
suffices (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z =>
(Gamma (2 * z))⁻¹ * (2 : ℂ) ^ (2 * z - 1) / ↑(Real.sqrt π) by
convert congr_arg Inv.inv (congr_fun this... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.574_0.in2QiCFW52coQT2 | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
h1 : AnalyticOn ℂ (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ
h2 : AnalyticOn ℂ (fun z => (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑(sqrt π)) univ
⊢ Tendsto ofReal' (𝓝[≠] 1) (𝓝[≠] 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [tendsto_nhdsWithin_iff] | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) := by
suffices (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z =>
(Gamma (2 * z))⁻¹ * (2 : ℂ) ^ (2 * z - 1) / ↑(Real.sqrt π) by
convert congr_arg Inv.inv (congr_fun this... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.574_0.in2QiCFW52coQT2 | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
h1 : AnalyticOn ℂ (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ
h2 : AnalyticOn ℂ (fun z => (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑(sqrt π)) univ
⊢ Tendsto ofReal' (𝓝[≠] 1) (𝓝 1) ∧ ∀ᶠ (n : ℝ) in 𝓝[≠] 1, ↑n ∈ {1}ᶜ | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | constructor | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) := by
suffices (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z =>
(Gamma (2 * z))⁻¹ * (2 : ℂ) ^ (2 * z - 1) / ↑(Real.sqrt π) by
convert congr_arg Inv.inv (congr_fun this... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.574_0.in2QiCFW52coQT2 | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case left
s : ℂ
h1 : AnalyticOn ℂ (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ
h2 : AnalyticOn ℂ (fun z => (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑(sqrt π)) univ
⊢ Tendsto ofReal' (𝓝[≠] 1) (𝓝 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | exact tendsto_nhdsWithin_of_tendsto_nhds continuous_ofReal.continuousAt | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) := by
suffices (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z =>
(Gamma (2 * z))⁻¹ * (2 : ℂ) ^ (2 * z - 1) / ↑(Real.sqrt π) by
convert congr_arg Inv.inv (congr_fun this... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.574_0.in2QiCFW52coQT2 | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case right
s : ℂ
h1 : AnalyticOn ℂ (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ
h2 : AnalyticOn ℂ (fun z => (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑(sqrt π)) univ
⊢ ∀ᶠ (n : ℝ) in 𝓝[≠] 1, ↑n ∈ {1}ᶜ | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | exact eventually_nhdsWithin_iff.mpr (eventually_of_forall fun t ht => ofReal_ne_one.mpr ht) | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) := by
suffices (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z =>
(Gamma (2 * z))⁻¹ * (2 : ℂ) ^ (2 * z - 1) / ↑(Real.sqrt π) by
convert congr_arg Inv.inv (congr_fun this... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.574_0.in2QiCFW52coQT2 | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
h1 : AnalyticOn ℂ (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ
h2 : AnalyticOn ℂ (fun z => (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑(sqrt π)) univ
h3 : Tendsto ofReal' (𝓝[≠] 1) (𝓝[≠] 1)
⊢ (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z => (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑(sqrt π) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | refine' AnalyticOn.eq_of_frequently_eq h1 h2 (h3.frequently _) | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) := by
suffices (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z =>
(Gamma (2 * z))⁻¹ * (2 : ℂ) ^ (2 * z - 1) / ↑(Real.sqrt π) by
convert congr_arg Inv.inv (congr_fun this... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.574_0.in2QiCFW52coQT2 | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
h1 : AnalyticOn ℂ (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ
h2 : AnalyticOn ℂ (fun z => (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑(sqrt π)) univ
h3 : Tendsto ofReal' (𝓝[≠] 1) (𝓝[≠] 1)
⊢ ∃ᶠ (x : ℝ) in 𝓝[≠] 1, (Gamma ↑x)⁻¹ * (Gamma (↑x + 1 / 2))⁻¹ = (Gamma (2 * ↑x))⁻¹ * 2 ^ (2 * ↑x - 1) / ↑(sqrt π) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | refine' ((Eventually.filter_mono nhdsWithin_le_nhds) _).frequently | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) := by
suffices (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z =>
(Gamma (2 * z))⁻¹ * (2 : ℂ) ^ (2 * z - 1) / ↑(Real.sqrt π) by
convert congr_arg Inv.inv (congr_fun this... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.574_0.in2QiCFW52coQT2 | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
h1 : AnalyticOn ℂ (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ
h2 : AnalyticOn ℂ (fun z => (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑(sqrt π)) univ
h3 : Tendsto ofReal' (𝓝[≠] 1) (𝓝[≠] 1)
⊢ ∀ᶠ (x : ℝ) in 𝓝 1, (Gamma ↑x)⁻¹ * (Gamma (↑x + 1 / 2))⁻¹ = (Gamma (2 * ↑x))⁻¹ * 2 ^ (2 * ↑x - 1) / ↑(sqrt π) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | refine' (eventually_gt_nhds zero_lt_one).mp (eventually_of_forall fun t ht => _) | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) := by
suffices (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z =>
(Gamma (2 * z))⁻¹ * (2 : ℂ) ^ (2 * z - 1) / ↑(Real.sqrt π) by
convert congr_arg Inv.inv (congr_fun this... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.574_0.in2QiCFW52coQT2 | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
h1 : AnalyticOn ℂ (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ
h2 : AnalyticOn ℂ (fun z => (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑(sqrt π)) univ
h3 : Tendsto ofReal' (𝓝[≠] 1) (𝓝[≠] 1)
t : ℝ
ht : 0 < t
⊢ (Gamma ↑t)⁻¹ * (Gamma (↑t + 1 / 2))⁻¹ = (Gamma (2 * ↑t))⁻¹ * 2 ^ (2 * ↑t - 1) / ↑(sqrt π) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [← mul_inv, Gamma_ofReal, (by norm_num : (t : ℂ) + 1 / 2 = ↑(t + 1 / 2)), Gamma_ofReal, ←
ofReal_mul, Gamma_mul_Gamma_add_half_of_pos ht, ofReal_mul, ofReal_mul, ← Gamma_ofReal,
mul_inv, mul_inv, (by norm_num : 2 * (t : ℂ) = ↑(2 * t)), Gamma_ofReal,
ofReal_cpow zero_le_two, show (2 : ℝ) = (2 : ℂ) by norm... | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) := by
suffices (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z =>
(Gamma (2 * z))⁻¹ * (2 : ℂ) ^ (2 * z - 1) / ↑(Real.sqrt π) by
convert congr_arg Inv.inv (congr_fun this... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.574_0.in2QiCFW52coQT2 | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
h1 : AnalyticOn ℂ (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ
h2 : AnalyticOn ℂ (fun z => (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑(sqrt π)) univ
h3 : Tendsto ofReal' (𝓝[≠] 1) (𝓝[≠] 1)
t : ℝ
ht : 0 < t
⊢ ↑t + 1 / 2 = ↑(t + 1 / 2) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | norm_num | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) := by
suffices (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z =>
(Gamma (2 * z))⁻¹ * (2 : ℂ) ^ (2 * z - 1) / ↑(Real.sqrt π) by
convert congr_arg Inv.inv (congr_fun this... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.574_0.in2QiCFW52coQT2 | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
h1 : AnalyticOn ℂ (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ
h2 : AnalyticOn ℂ (fun z => (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑(sqrt π)) univ
h3 : Tendsto ofReal' (𝓝[≠] 1) (𝓝[≠] 1)
t : ℝ
ht : 0 < t
⊢ 2 * ↑t = ↑(2 * t) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | norm_num | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) := by
suffices (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z =>
(Gamma (2 * z))⁻¹ * (2 : ℂ) ^ (2 * z - 1) / ↑(Real.sqrt π) by
convert congr_arg Inv.inv (congr_fun this... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.574_0.in2QiCFW52coQT2 | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
h1 : AnalyticOn ℂ (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ
h2 : AnalyticOn ℂ (fun z => (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑(sqrt π)) univ
h3 : Tendsto ofReal' (𝓝[≠] 1) (𝓝[≠] 1)
t : ℝ
ht : 0 < t
⊢ ↑2 = 2 | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | norm_cast | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) := by
suffices (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z =>
(Gamma (2 * z))⁻¹ * (2 : ℂ) ^ (2 * z - 1) / ↑(Real.sqrt π) by
convert congr_arg Inv.inv (congr_fun this... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.574_0.in2QiCFW52coQT2 | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(Real.sqrt π) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℝ
⊢ Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt π | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [← ofReal_inj] | theorem Gamma_mul_Gamma_add_half (s : ℝ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℝ) ^ (1 - 2 * s) * sqrt π := by
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.613_0.in2QiCFW52coQT2 | theorem Gamma_mul_Gamma_add_half (s : ℝ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℝ) ^ (1 - 2 * s) * sqrt π | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℝ
⊢ ↑(Gamma s * Gamma (s + 1 / 2)) = ↑(Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt π) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | simpa only [← Gamma_ofReal, ofReal_cpow zero_le_two, ofReal_mul, ofReal_add, ofReal_div,
ofReal_sub] using Complex.Gamma_mul_Gamma_add_half ↑s | theorem Gamma_mul_Gamma_add_half (s : ℝ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℝ) ^ (1 - 2 * s) * sqrt π := by
rw [← ofReal_inj]
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.613_0.in2QiCFW52coQT2 | theorem Gamma_mul_Gamma_add_half (s : ℝ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℝ) ^ (1 - 2 * s) * sqrt π | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ Measure.HaveLebesgueDecomposition (toJordanDecomposition (-s)).posPart μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [toJordanDecomposition_neg, JordanDecomposition.neg_posPart] | instance haveLebesgueDecomposition_neg (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] : (-s).HaveLebesgueDecomposition μ where
posPart := by
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.84_0.HPGboz0rhL6sBes | instance haveLebesgueDecomposition_neg (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] : (-s).HaveLebesgueDecomposition μ where
posPart | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ Measure.HaveLebesgueDecomposition (toJordanDecomposition s).negPart μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | infer_instance | instance haveLebesgueDecomposition_neg (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] : (-s).HaveLebesgueDecomposition μ where
posPart := by
rw [toJordanDecomposition_neg, JordanDecomposition.neg_posPart]
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.84_0.HPGboz0rhL6sBes | instance haveLebesgueDecomposition_neg (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] : (-s).HaveLebesgueDecomposition μ where
posPart | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ Measure.HaveLebesgueDecomposition (toJordanDecomposition (-s)).negPart μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [toJordanDecomposition_neg, JordanDecomposition.neg_negPart] | instance haveLebesgueDecomposition_neg (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] : (-s).HaveLebesgueDecomposition μ where
posPart := by
rw [toJordanDecomposition_neg, JordanDecomposition.neg_posPart]
infer_instance
negPart := by
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.84_0.HPGboz0rhL6sBes | instance haveLebesgueDecomposition_neg (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] : (-s).HaveLebesgueDecomposition μ where
posPart | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ Measure.HaveLebesgueDecomposition (toJordanDecomposition s).posPart μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | infer_instance | instance haveLebesgueDecomposition_neg (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] : (-s).HaveLebesgueDecomposition μ where
posPart := by
rw [toJordanDecomposition_neg, JordanDecomposition.neg_posPart]
infer_instance
negPart := by
rw [toJordanDecomposition_neg, JordanDecomp... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.84_0.HPGboz0rhL6sBes | instance haveLebesgueDecomposition_neg (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] : (-s).HaveLebesgueDecomposition μ where
posPart | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
inst✝ : HaveLebesgueDecomposition s μ
r : ℝ≥0
⊢ Measure.HaveLebesgueDecomposition (toJordanDecomposition (r • s)).posPart μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [toJordanDecomposition_smul, JordanDecomposition.smul_posPart] | instance haveLebesgueDecomposition_smul (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] (r : ℝ≥0) : (r • s).HaveLebesgueDecomposition μ where
posPart := by
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.94_0.HPGboz0rhL6sBes | instance haveLebesgueDecomposition_smul (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] (r : ℝ≥0) : (r • s).HaveLebesgueDecomposition μ where
posPart | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
inst✝ : HaveLebesgueDecomposition s μ
r : ℝ≥0
⊢ Measure.HaveLebesgueDecomposition (r • (toJordanDecomposition s).posPart) μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | infer_instance | instance haveLebesgueDecomposition_smul (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] (r : ℝ≥0) : (r • s).HaveLebesgueDecomposition μ where
posPart := by
rw [toJordanDecomposition_smul, JordanDecomposition.smul_posPart]
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.94_0.HPGboz0rhL6sBes | instance haveLebesgueDecomposition_smul (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] (r : ℝ≥0) : (r • s).HaveLebesgueDecomposition μ where
posPart | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
inst✝ : HaveLebesgueDecomposition s μ
r : ℝ≥0
⊢ Measure.HaveLebesgueDecomposition (toJordanDecomposition (r • s)).negPart μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [toJordanDecomposition_smul, JordanDecomposition.smul_negPart] | instance haveLebesgueDecomposition_smul (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] (r : ℝ≥0) : (r • s).HaveLebesgueDecomposition μ where
posPart := by
rw [toJordanDecomposition_smul, JordanDecomposition.smul_posPart]
infer_instance
negPart := by
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.94_0.HPGboz0rhL6sBes | instance haveLebesgueDecomposition_smul (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] (r : ℝ≥0) : (r • s).HaveLebesgueDecomposition μ where
posPart | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
inst✝ : HaveLebesgueDecomposition s μ
r : ℝ≥0
⊢ Measure.HaveLebesgueDecomposition (r • (toJordanDecomposition s).negPart) μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | infer_instance | instance haveLebesgueDecomposition_smul (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] (r : ℝ≥0) : (r • s).HaveLebesgueDecomposition μ where
posPart := by
rw [toJordanDecomposition_smul, JordanDecomposition.smul_posPart]
infer_instance
negPart := by
rw [toJordanDecomposition_s... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.94_0.HPGboz0rhL6sBes | instance haveLebesgueDecomposition_smul (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] (r : ℝ≥0) : (r • s).HaveLebesgueDecomposition μ where
posPart | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
inst✝ : HaveLebesgueDecomposition s μ
r : ℝ
⊢ HaveLebesgueDecomposition (r • s) μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | by_cases hr : 0 ≤ r | instance haveLebesgueDecomposition_smul_real (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] (r : ℝ) : (r • s).HaveLebesgueDecomposition μ := by
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.104_0.HPGboz0rhL6sBes | instance haveLebesgueDecomposition_smul_real (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] (r : ℝ) : (r • s).HaveLebesgueDecomposition μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
inst✝ : HaveLebesgueDecomposition s μ
r : ℝ
hr : 0 ≤ r
⊢ HaveLebesgueDecomposition (r • s) μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | lift r to ℝ≥0 using hr | instance haveLebesgueDecomposition_smul_real (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] (r : ℝ) : (r • s).HaveLebesgueDecomposition μ := by
by_cases hr : 0 ≤ r
· | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.104_0.HPGboz0rhL6sBes | instance haveLebesgueDecomposition_smul_real (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] (r : ℝ) : (r • s).HaveLebesgueDecomposition μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case pos.intro
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
inst✝ : HaveLebesgueDecomposition s μ
r : ℝ≥0
⊢ HaveLebesgueDecomposition (↑r • s) μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | exact s.haveLebesgueDecomposition_smul μ _ | instance haveLebesgueDecomposition_smul_real (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] (r : ℝ) : (r • s).HaveLebesgueDecomposition μ := by
by_cases hr : 0 ≤ r
· lift r to ℝ≥0 using hr
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.104_0.HPGboz0rhL6sBes | instance haveLebesgueDecomposition_smul_real (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] (r : ℝ) : (r • s).HaveLebesgueDecomposition μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
inst✝ : HaveLebesgueDecomposition s μ
r : ℝ
hr : ¬0 ≤ r
⊢ HaveLebesgueDecomposition (r • s) μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [not_le] at hr | instance haveLebesgueDecomposition_smul_real (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] (r : ℝ) : (r • s).HaveLebesgueDecomposition μ := by
by_cases hr : 0 ≤ r
· lift r to ℝ≥0 using hr
exact s.haveLebesgueDecomposition_smul μ _
· | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.104_0.HPGboz0rhL6sBes | instance haveLebesgueDecomposition_smul_real (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] (r : ℝ) : (r • s).HaveLebesgueDecomposition μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
inst✝ : HaveLebesgueDecomposition s μ
r : ℝ
hr : r < 0
⊢ HaveLebesgueDecomposition (r • s) μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | refine'
{ posPart := by
rw [toJordanDecomposition_smul_real, JordanDecomposition.real_smul_posPart_neg _ _ hr]
infer_instance
negPart := by
rw [toJordanDecomposition_smul_real, JordanDecomposition.real_smul_negPart_neg _ _ hr]
infer_instance } | instance haveLebesgueDecomposition_smul_real (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] (r : ℝ) : (r • s).HaveLebesgueDecomposition μ := by
by_cases hr : 0 ≤ r
· lift r to ℝ≥0 using hr
exact s.haveLebesgueDecomposition_smul μ _
· rw [not_le] at hr
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.104_0.HPGboz0rhL6sBes | instance haveLebesgueDecomposition_smul_real (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] (r : ℝ) : (r • s).HaveLebesgueDecomposition μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
inst✝ : HaveLebesgueDecomposition s μ
r : ℝ
hr : r < 0
⊢ Measure.HaveLebesgueDecomposition (toJordanDecomposition (r • s)).posPart μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [toJordanDecomposition_smul_real, JordanDecomposition.real_smul_posPart_neg _ _ hr] | instance haveLebesgueDecomposition_smul_real (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] (r : ℝ) : (r • s).HaveLebesgueDecomposition μ := by
by_cases hr : 0 ≤ r
· lift r to ℝ≥0 using hr
exact s.haveLebesgueDecomposition_smul μ _
· rw [not_le] at hr
refine'
{ posPart := ... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.104_0.HPGboz0rhL6sBes | instance haveLebesgueDecomposition_smul_real (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] (r : ℝ) : (r • s).HaveLebesgueDecomposition μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
inst✝ : HaveLebesgueDecomposition s μ
r : ℝ
hr : r < 0
⊢ Measure.HaveLebesgueDecomposition (Real.toNNReal (-r) • (toJordanDecomposition s).negPart) μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | infer_instance | instance haveLebesgueDecomposition_smul_real (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] (r : ℝ) : (r • s).HaveLebesgueDecomposition μ := by
by_cases hr : 0 ≤ r
· lift r to ℝ≥0 using hr
exact s.haveLebesgueDecomposition_smul μ _
· rw [not_le] at hr
refine'
{ posPart := ... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.104_0.HPGboz0rhL6sBes | instance haveLebesgueDecomposition_smul_real (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] (r : ℝ) : (r • s).HaveLebesgueDecomposition μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
inst✝ : HaveLebesgueDecomposition s μ
r : ℝ
hr : r < 0
⊢ Measure.HaveLebesgueDecomposition (toJordanDecomposition (r • s)).negPart μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [toJordanDecomposition_smul_real, JordanDecomposition.real_smul_negPart_neg _ _ hr] | instance haveLebesgueDecomposition_smul_real (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] (r : ℝ) : (r • s).HaveLebesgueDecomposition μ := by
by_cases hr : 0 ≤ r
· lift r to ℝ≥0 using hr
exact s.haveLebesgueDecomposition_smul μ _
· rw [not_le] at hr
refine'
{ posPart := ... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.104_0.HPGboz0rhL6sBes | instance haveLebesgueDecomposition_smul_real (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] (r : ℝ) : (r • s).HaveLebesgueDecomposition μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
inst✝ : HaveLebesgueDecomposition s μ
r : ℝ
hr : r < 0
⊢ Measure.HaveLebesgueDecomposition (Real.toNNReal (-r) • (toJordanDecomposition s).posPart) μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | infer_instance | instance haveLebesgueDecomposition_smul_real (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] (r : ℝ) : (r • s).HaveLebesgueDecomposition μ := by
by_cases hr : 0 ≤ r
· lift r to ℝ≥0 using hr
exact s.haveLebesgueDecomposition_smul μ _
· rw [not_le] at hr
refine'
{ posPart := ... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.104_0.HPGboz0rhL6sBes | instance haveLebesgueDecomposition_smul_real (s : SignedMeasure α) (μ : Measure α)
[s.HaveLebesgueDecomposition μ] (r : ℝ) : (r • s).HaveLebesgueDecomposition μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
⊢ Measure.singularPart (toJordanDecomposition s).posPart μ ⟂ₘ Measure.singularPart (toJordanDecomposition s).negPart μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | by_cases hl : s.HaveLebesgueDecomposition μ | theorem singularPart_mutuallySingular (s : SignedMeasure α) (μ : Measure α) :
s.toJordanDecomposition.posPart.singularPart μ ⟂ₘ
s.toJordanDecomposition.negPart.singularPart μ := by
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.129_0.HPGboz0rhL6sBes | theorem singularPart_mutuallySingular (s : SignedMeasure α) (μ : Measure α) :
s.toJordanDecomposition.posPart.singularPart μ ⟂ₘ
s.toJordanDecomposition.negPart.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
hl : HaveLebesgueDecomposition s μ
⊢ Measure.singularPart (toJordanDecomposition s).posPart μ ⟂ₘ Measure.singularPart (toJordanDecomposition s).negPart μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | obtain ⟨i, hi, hpos, hneg⟩ := s.toJordanDecomposition.mutuallySingular | theorem singularPart_mutuallySingular (s : SignedMeasure α) (μ : Measure α) :
s.toJordanDecomposition.posPart.singularPart μ ⟂ₘ
s.toJordanDecomposition.negPart.singularPart μ := by
by_cases hl : s.HaveLebesgueDecomposition μ
· | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.129_0.HPGboz0rhL6sBes | theorem singularPart_mutuallySingular (s : SignedMeasure α) (μ : Measure α) :
s.toJordanDecomposition.posPart.singularPart μ ⟂ₘ
s.toJordanDecomposition.negPart.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case pos.intro.intro.intro
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
hl : HaveLebesgueDecomposition s μ
i : Set α
hi : MeasurableSet i
hpos : ↑↑(toJordanDecomposition s).posPart i = 0
hneg : ↑↑(toJordanDecomposition s).negPart iᶜ = 0
⊢ Measure.singularPart (toJor... | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [s.toJordanDecomposition.posPart.haveLebesgueDecomposition_add μ] at hpos | theorem singularPart_mutuallySingular (s : SignedMeasure α) (μ : Measure α) :
s.toJordanDecomposition.posPart.singularPart μ ⟂ₘ
s.toJordanDecomposition.negPart.singularPart μ := by
by_cases hl : s.HaveLebesgueDecomposition μ
· obtain ⟨i, hi, hpos, hneg⟩ := s.toJordanDecomposition.mutuallySingular
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.129_0.HPGboz0rhL6sBes | theorem singularPart_mutuallySingular (s : SignedMeasure α) (μ : Measure α) :
s.toJordanDecomposition.posPart.singularPart μ ⟂ₘ
s.toJordanDecomposition.negPart.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case pos.intro.intro.intro
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
hl : HaveLebesgueDecomposition s μ
i : Set α
hi : MeasurableSet i
hpos :
↑↑(Measure.singularPart (toJordanDecomposition s).posPart μ +
withDensity μ (rnDeriv (toJordanDecomposition... | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [s.toJordanDecomposition.negPart.haveLebesgueDecomposition_add μ] at hneg | theorem singularPart_mutuallySingular (s : SignedMeasure α) (μ : Measure α) :
s.toJordanDecomposition.posPart.singularPart μ ⟂ₘ
s.toJordanDecomposition.negPart.singularPart μ := by
by_cases hl : s.HaveLebesgueDecomposition μ
· obtain ⟨i, hi, hpos, hneg⟩ := s.toJordanDecomposition.mutuallySingular
rw [... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.129_0.HPGboz0rhL6sBes | theorem singularPart_mutuallySingular (s : SignedMeasure α) (μ : Measure α) :
s.toJordanDecomposition.posPart.singularPart μ ⟂ₘ
s.toJordanDecomposition.negPart.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case pos.intro.intro.intro
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
hl : HaveLebesgueDecomposition s μ
i : Set α
hi : MeasurableSet i
hpos :
↑↑(Measure.singularPart (toJordanDecomposition s).posPart μ +
withDensity μ (rnDeriv (toJordanDecomposition... | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [add_apply, add_eq_zero_iff] at hpos hneg | theorem singularPart_mutuallySingular (s : SignedMeasure α) (μ : Measure α) :
s.toJordanDecomposition.posPart.singularPart μ ⟂ₘ
s.toJordanDecomposition.negPart.singularPart μ := by
by_cases hl : s.HaveLebesgueDecomposition μ
· obtain ⟨i, hi, hpos, hneg⟩ := s.toJordanDecomposition.mutuallySingular
rw [... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.129_0.HPGboz0rhL6sBes | theorem singularPart_mutuallySingular (s : SignedMeasure α) (μ : Measure α) :
s.toJordanDecomposition.posPart.singularPart μ ⟂ₘ
s.toJordanDecomposition.negPart.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case pos.intro.intro.intro
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
hl : HaveLebesgueDecomposition s μ
i : Set α
hi : MeasurableSet i
hpos :
↑↑(Measure.singularPart (toJordanDecomposition s).posPart μ) i = 0 ∧
↑↑(withDensity μ (rnDeriv (toJordanDecompositi... | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | exact ⟨i, hi, hpos.1, hneg.1⟩ | theorem singularPart_mutuallySingular (s : SignedMeasure α) (μ : Measure α) :
s.toJordanDecomposition.posPart.singularPart μ ⟂ₘ
s.toJordanDecomposition.negPart.singularPart μ := by
by_cases hl : s.HaveLebesgueDecomposition μ
· obtain ⟨i, hi, hpos, hneg⟩ := s.toJordanDecomposition.mutuallySingular
rw [... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.129_0.HPGboz0rhL6sBes | theorem singularPart_mutuallySingular (s : SignedMeasure α) (μ : Measure α) :
s.toJordanDecomposition.posPart.singularPart μ ⟂ₘ
s.toJordanDecomposition.negPart.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
hl : ¬HaveLebesgueDecomposition s μ
⊢ Measure.singularPart (toJordanDecomposition s).posPart μ ⟂ₘ Measure.singularPart (toJordanDecomposition s).negPart μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [not_haveLebesgueDecomposition_iff] at hl | theorem singularPart_mutuallySingular (s : SignedMeasure α) (μ : Measure α) :
s.toJordanDecomposition.posPart.singularPart μ ⟂ₘ
s.toJordanDecomposition.negPart.singularPart μ := by
by_cases hl : s.HaveLebesgueDecomposition μ
· obtain ⟨i, hi, hpos, hneg⟩ := s.toJordanDecomposition.mutuallySingular
rw [... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.129_0.HPGboz0rhL6sBes | theorem singularPart_mutuallySingular (s : SignedMeasure α) (μ : Measure α) :
s.toJordanDecomposition.posPart.singularPart μ ⟂ₘ
s.toJordanDecomposition.negPart.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
hl :
¬Measure.HaveLebesgueDecomposition (toJordanDecomposition s).posPart μ ∨
¬Measure.HaveLebesgueDecomposition (toJordanDecomposition s).negPart μ
⊢ Measure.singularPart (toJordanDecomposition s).posPart ... | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | cases' hl with hp hn | theorem singularPart_mutuallySingular (s : SignedMeasure α) (μ : Measure α) :
s.toJordanDecomposition.posPart.singularPart μ ⟂ₘ
s.toJordanDecomposition.negPart.singularPart μ := by
by_cases hl : s.HaveLebesgueDecomposition μ
· obtain ⟨i, hi, hpos, hneg⟩ := s.toJordanDecomposition.mutuallySingular
rw [... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.129_0.HPGboz0rhL6sBes | theorem singularPart_mutuallySingular (s : SignedMeasure α) (μ : Measure α) :
s.toJordanDecomposition.posPart.singularPart μ ⟂ₘ
s.toJordanDecomposition.negPart.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case neg.inl
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
hp : ¬Measure.HaveLebesgueDecomposition (toJordanDecomposition s).posPart μ
⊢ Measure.singularPart (toJordanDecomposition s).posPart μ ⟂ₘ Measure.singularPart (toJordanDecomposition s).negPart μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [Measure.singularPart, dif_neg hp] | theorem singularPart_mutuallySingular (s : SignedMeasure α) (μ : Measure α) :
s.toJordanDecomposition.posPart.singularPart μ ⟂ₘ
s.toJordanDecomposition.negPart.singularPart μ := by
by_cases hl : s.HaveLebesgueDecomposition μ
· obtain ⟨i, hi, hpos, hneg⟩ := s.toJordanDecomposition.mutuallySingular
rw [... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.129_0.HPGboz0rhL6sBes | theorem singularPart_mutuallySingular (s : SignedMeasure α) (μ : Measure α) :
s.toJordanDecomposition.posPart.singularPart μ ⟂ₘ
s.toJordanDecomposition.negPart.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case neg.inl
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
hp : ¬Measure.HaveLebesgueDecomposition (toJordanDecomposition s).posPart μ
⊢ 0 ⟂ₘ Measure.singularPart (toJordanDecomposition s).negPart μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | exact MutuallySingular.zero_left | theorem singularPart_mutuallySingular (s : SignedMeasure α) (μ : Measure α) :
s.toJordanDecomposition.posPart.singularPart μ ⟂ₘ
s.toJordanDecomposition.negPart.singularPart μ := by
by_cases hl : s.HaveLebesgueDecomposition μ
· obtain ⟨i, hi, hpos, hneg⟩ := s.toJordanDecomposition.mutuallySingular
rw [... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.129_0.HPGboz0rhL6sBes | theorem singularPart_mutuallySingular (s : SignedMeasure α) (μ : Measure α) :
s.toJordanDecomposition.posPart.singularPart μ ⟂ₘ
s.toJordanDecomposition.negPart.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case neg.inr
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
hn : ¬Measure.HaveLebesgueDecomposition (toJordanDecomposition s).negPart μ
⊢ Measure.singularPart (toJordanDecomposition s).posPart μ ⟂ₘ Measure.singularPart (toJordanDecomposition s).negPart μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [Measure.singularPart, Measure.singularPart, dif_neg hn] | theorem singularPart_mutuallySingular (s : SignedMeasure α) (μ : Measure α) :
s.toJordanDecomposition.posPart.singularPart μ ⟂ₘ
s.toJordanDecomposition.negPart.singularPart μ := by
by_cases hl : s.HaveLebesgueDecomposition μ
· obtain ⟨i, hi, hpos, hneg⟩ := s.toJordanDecomposition.mutuallySingular
rw [... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.129_0.HPGboz0rhL6sBes | theorem singularPart_mutuallySingular (s : SignedMeasure α) (μ : Measure α) :
s.toJordanDecomposition.posPart.singularPart μ ⟂ₘ
s.toJordanDecomposition.negPart.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case neg.inr
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
hn : ¬Measure.HaveLebesgueDecomposition (toJordanDecomposition s).negPart μ
⊢ (if h : Measure.HaveLebesgueDecomposition (toJordanDecomposition s).posPart μ then
(Classical.choose
(_ : ∃ p, Mea... | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | exact MutuallySingular.zero_right | theorem singularPart_mutuallySingular (s : SignedMeasure α) (μ : Measure α) :
s.toJordanDecomposition.posPart.singularPart μ ⟂ₘ
s.toJordanDecomposition.negPart.singularPart μ := by
by_cases hl : s.HaveLebesgueDecomposition μ
· obtain ⟨i, hi, hpos, hneg⟩ := s.toJordanDecomposition.mutuallySingular
rw [... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.129_0.HPGboz0rhL6sBes | theorem singularPart_mutuallySingular (s : SignedMeasure α) (μ : Measure α) :
s.toJordanDecomposition.posPart.singularPart μ ⟂ₘ
s.toJordanDecomposition.negPart.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
⊢ totalVariation (singularPart s μ) =
Measure.singularPart (toJordanDecomposition s).posPart μ + Measure.singularPart (toJordanDecomposition s).negPart μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | have :
(s.singularPart μ).toJordanDecomposition =
⟨s.toJordanDecomposition.posPart.singularPart μ,
s.toJordanDecomposition.negPart.singularPart μ, singularPart_mutuallySingular s μ⟩ := by
refine' JordanDecomposition.toSignedMeasure_injective _
rw [toSignedMeasure_toJordanDecomposition, singula... | theorem singularPart_totalVariation (s : SignedMeasure α) (μ : Measure α) :
(s.singularPart μ).totalVariation =
s.toJordanDecomposition.posPart.singularPart μ +
s.toJordanDecomposition.negPart.singularPart μ := by
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.146_0.HPGboz0rhL6sBes | theorem singularPart_totalVariation (s : SignedMeasure α) (μ : Measure α) :
(s.singularPart μ).totalVariation =
s.toJordanDecomposition.posPart.singularPart μ +
s.toJordanDecomposition.negPart.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
⊢ toJordanDecomposition (singularPart s μ) =
JordanDecomposition.mk (Measure.singularPart (toJordanDecomposition s).posPart μ)
(Measure.singularPart (toJordanDecomposition s).negPart μ)
(_ :
Measur... | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | refine' JordanDecomposition.toSignedMeasure_injective _ | theorem singularPart_totalVariation (s : SignedMeasure α) (μ : Measure α) :
(s.singularPart μ).totalVariation =
s.toJordanDecomposition.posPart.singularPart μ +
s.toJordanDecomposition.negPart.singularPart μ := by
have :
(s.singularPart μ).toJordanDecomposition =
⟨s.toJordanDecomposition.p... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.146_0.HPGboz0rhL6sBes | theorem singularPart_totalVariation (s : SignedMeasure α) (μ : Measure α) :
(s.singularPart μ).totalVariation =
s.toJordanDecomposition.posPart.singularPart μ +
s.toJordanDecomposition.negPart.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
⊢ JordanDecomposition.toSignedMeasure (toJordanDecomposition (singularPart s μ)) =
JordanDecomposition.toSignedMeasure
(JordanDecomposition.mk (Measure.singularPart (toJordanDecomposition s).posPart μ)
(... | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [toSignedMeasure_toJordanDecomposition, singularPart, JordanDecomposition.toSignedMeasure] | theorem singularPart_totalVariation (s : SignedMeasure α) (μ : Measure α) :
(s.singularPart μ).totalVariation =
s.toJordanDecomposition.posPart.singularPart μ +
s.toJordanDecomposition.negPart.singularPart μ := by
have :
(s.singularPart μ).toJordanDecomposition =
⟨s.toJordanDecomposition.p... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.146_0.HPGboz0rhL6sBes | theorem singularPart_totalVariation (s : SignedMeasure α) (μ : Measure α) :
(s.singularPart μ).totalVariation =
s.toJordanDecomposition.posPart.singularPart μ +
s.toJordanDecomposition.negPart.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
this :
toJordanDecomposition (singularPart s μ) =
JordanDecomposition.mk (Measure.singularPart (toJordanDecomposition s).posPart μ)
(Measure.singularPart (toJordanDecomposition s).negPart μ)
(_ :
... | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [totalVariation, this] | theorem singularPart_totalVariation (s : SignedMeasure α) (μ : Measure α) :
(s.singularPart μ).totalVariation =
s.toJordanDecomposition.posPart.singularPart μ +
s.toJordanDecomposition.negPart.singularPart μ := by
have :
(s.singularPart μ).toJordanDecomposition =
⟨s.toJordanDecomposition.p... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.146_0.HPGboz0rhL6sBes | theorem singularPart_totalVariation (s : SignedMeasure α) (μ : Measure α) :
(s.singularPart μ).totalVariation =
s.toJordanDecomposition.posPart.singularPart μ +
s.toJordanDecomposition.negPart.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
⊢ singularPart s μ ⟂ᵥ toENNRealVectorMeasure μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [mutuallySingular_ennreal_iff, singularPart_totalVariation,
VectorMeasure.ennrealToMeasure_toENNRealVectorMeasure] | nonrec theorem mutuallySingular_singularPart (s : SignedMeasure α) (μ : Measure α) :
singularPart s μ ⟂ᵥ μ.toENNRealVectorMeasure := by
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.159_0.HPGboz0rhL6sBes | nonrec theorem mutuallySingular_singularPart (s : SignedMeasure α) (μ : Measure α) :
singularPart s μ ⟂ᵥ μ.toENNRealVectorMeasure | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s : SignedMeasure α
μ : Measure α
⊢ Measure.singularPart (toJordanDecomposition s).posPart μ + Measure.singularPart (toJordanDecomposition s).negPart μ ⟂ₘ
μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | exact (mutuallySingular_singularPart _ _).add_left (mutuallySingular_singularPart _ _) | nonrec theorem mutuallySingular_singularPart (s : SignedMeasure α) (μ : Measure α) :
singularPart s μ ⟂ᵥ μ.toENNRealVectorMeasure := by
rw [mutuallySingular_ennreal_iff, singularPart_totalVariation,
VectorMeasure.ennrealToMeasure_toENNRealVectorMeasure]
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.159_0.HPGboz0rhL6sBes | nonrec theorem mutuallySingular_singularPart (s : SignedMeasure α) (μ : Measure α) :
singularPart s μ ⟂ᵥ μ.toENNRealVectorMeasure | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
⊢ Measurable (rnDeriv s μ) | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [rnDeriv] | @[measurability]
theorem measurable_rnDeriv (s : SignedMeasure α) (μ : Measure α) : Measurable (rnDeriv s μ) := by
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.190_0.HPGboz0rhL6sBes | @[measurability]
theorem measurable_rnDeriv (s : SignedMeasure α) (μ : Measure α) : Measurable (rnDeriv s μ) | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
⊢ Measurable fun x =>
ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).posPart μ x) -
ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).negPart μ x) | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | measurability | @[measurability]
theorem measurable_rnDeriv (s : SignedMeasure α) (μ : Measure α) : Measurable (rnDeriv s μ) := by
rw [rnDeriv]
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.190_0.HPGboz0rhL6sBes | @[measurability]
theorem measurable_rnDeriv (s : SignedMeasure α) (μ : Measure α) : Measurable (rnDeriv s μ) | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
⊢ Integrable (rnDeriv s μ) | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | refine' Integrable.sub _ _ | theorem integrable_rnDeriv (s : SignedMeasure α) (μ : Measure α) : Integrable (rnDeriv s μ) μ := by
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.196_0.HPGboz0rhL6sBes | theorem integrable_rnDeriv (s : SignedMeasure α) (μ : Measure α) : Integrable (rnDeriv s μ) μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case refine'_1
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
⊢ Integrable fun x => ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).posPart μ x) | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | constructor | theorem integrable_rnDeriv (s : SignedMeasure α) (μ : Measure α) : Integrable (rnDeriv s μ) μ := by
refine' Integrable.sub _ _ <;>
· | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.196_0.HPGboz0rhL6sBes | theorem integrable_rnDeriv (s : SignedMeasure α) (μ : Measure α) : Integrable (rnDeriv s μ) μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case refine'_1.left
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
⊢ AEStronglyMeasurable (fun x => ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).posPart μ x)) μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | apply Measurable.aestronglyMeasurable | theorem integrable_rnDeriv (s : SignedMeasure α) (μ : Measure α) : Integrable (rnDeriv s μ) μ := by
refine' Integrable.sub _ _ <;>
· constructor
· | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.196_0.HPGboz0rhL6sBes | theorem integrable_rnDeriv (s : SignedMeasure α) (μ : Measure α) : Integrable (rnDeriv s μ) μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case refine'_1.left.hf
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
⊢ Measurable fun x => ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).posPart μ x) | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | measurability | theorem integrable_rnDeriv (s : SignedMeasure α) (μ : Measure α) : Integrable (rnDeriv s μ) μ := by
refine' Integrable.sub _ _ <;>
· constructor
· apply Measurable.aestronglyMeasurable; | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.196_0.HPGboz0rhL6sBes | theorem integrable_rnDeriv (s : SignedMeasure α) (μ : Measure α) : Integrable (rnDeriv s μ) μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case refine'_1.right
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
⊢ HasFiniteIntegral fun x => ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).posPart μ x) | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | exact hasFiniteIntegral_toReal_of_lintegral_ne_top (lintegral_rnDeriv_lt_top _ μ).ne | theorem integrable_rnDeriv (s : SignedMeasure α) (μ : Measure α) : Integrable (rnDeriv s μ) μ := by
refine' Integrable.sub _ _ <;>
· constructor
· apply Measurable.aestronglyMeasurable; measurability
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.196_0.HPGboz0rhL6sBes | theorem integrable_rnDeriv (s : SignedMeasure α) (μ : Measure α) : Integrable (rnDeriv s μ) μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case refine'_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
⊢ Integrable fun x => ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).negPart μ x) | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | constructor | theorem integrable_rnDeriv (s : SignedMeasure α) (μ : Measure α) : Integrable (rnDeriv s μ) μ := by
refine' Integrable.sub _ _ <;>
· | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.196_0.HPGboz0rhL6sBes | theorem integrable_rnDeriv (s : SignedMeasure α) (μ : Measure α) : Integrable (rnDeriv s μ) μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case refine'_2.left
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
⊢ AEStronglyMeasurable (fun x => ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).negPart μ x)) μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | apply Measurable.aestronglyMeasurable | theorem integrable_rnDeriv (s : SignedMeasure α) (μ : Measure α) : Integrable (rnDeriv s μ) μ := by
refine' Integrable.sub _ _ <;>
· constructor
· | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.196_0.HPGboz0rhL6sBes | theorem integrable_rnDeriv (s : SignedMeasure α) (μ : Measure α) : Integrable (rnDeriv s μ) μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case refine'_2.left.hf
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
⊢ Measurable fun x => ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).negPart μ x) | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | measurability | theorem integrable_rnDeriv (s : SignedMeasure α) (μ : Measure α) : Integrable (rnDeriv s μ) μ := by
refine' Integrable.sub _ _ <;>
· constructor
· apply Measurable.aestronglyMeasurable; | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.196_0.HPGboz0rhL6sBes | theorem integrable_rnDeriv (s : SignedMeasure α) (μ : Measure α) : Integrable (rnDeriv s μ) μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case refine'_2.right
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
⊢ HasFiniteIntegral fun x => ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).negPart μ x) | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | exact hasFiniteIntegral_toReal_of_lintegral_ne_top (lintegral_rnDeriv_lt_top _ μ).ne | theorem integrable_rnDeriv (s : SignedMeasure α) (μ : Measure α) : Integrable (rnDeriv s μ) μ := by
refine' Integrable.sub _ _ <;>
· constructor
· apply Measurable.aestronglyMeasurable; measurability
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.196_0.HPGboz0rhL6sBes | theorem integrable_rnDeriv (s : SignedMeasure α) (μ : Measure α) : Integrable (rnDeriv s μ) μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ singularPart s μ + withDensityᵥ μ (rnDeriv s μ) = s | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | conv_rhs =>
rw [← toSignedMeasure_toJordanDecomposition s, JordanDecomposition.toSignedMeasure] | /-- **The Lebesgue Decomposition theorem between a signed measure and a measure**:
Given a signed measure `s` and a σ-finite measure `μ`, there exist a signed measure `t` and a
measurable and integrable function `f`, such that `t` is mutually singular with respect to `μ`
and `s = t + μ.withDensityᵥ f`. In this case `t ... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.205_0.HPGboz0rhL6sBes | /-- **The Lebesgue Decomposition theorem between a signed measure and a measure**:
Given a signed measure `s` and a σ-finite measure `μ`, there exist a signed measure `t` and a
measurable and integrable function `f`, such that `t` is mutually singular with respect to `μ`
and `s = t + μ.withDensityᵥ f`. In this case `t ... | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
| s | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [← toSignedMeasure_toJordanDecomposition s, JordanDecomposition.toSignedMeasure] | /-- **The Lebesgue Decomposition theorem between a signed measure and a measure**:
Given a signed measure `s` and a σ-finite measure `μ`, there exist a signed measure `t` and a
measurable and integrable function `f`, such that `t` is mutually singular with respect to `μ`
and `s = t + μ.withDensityᵥ f`. In this case `t ... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.205_0.HPGboz0rhL6sBes | /-- **The Lebesgue Decomposition theorem between a signed measure and a measure**:
Given a signed measure `s` and a σ-finite measure `μ`, there exist a signed measure `t` and a
measurable and integrable function `f`, such that `t` is mutually singular with respect to `μ`
and `s = t + μ.withDensityᵥ f`. In this case `t ... | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
| s | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [← toSignedMeasure_toJordanDecomposition s, JordanDecomposition.toSignedMeasure] | /-- **The Lebesgue Decomposition theorem between a signed measure and a measure**:
Given a signed measure `s` and a σ-finite measure `μ`, there exist a signed measure `t` and a
measurable and integrable function `f`, such that `t` is mutually singular with respect to `μ`
and `s = t + μ.withDensityᵥ f`. In this case `t ... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.205_0.HPGboz0rhL6sBes | /-- **The Lebesgue Decomposition theorem between a signed measure and a measure**:
Given a signed measure `s` and a σ-finite measure `μ`, there exist a signed measure `t` and a
measurable and integrable function `f`, such that `t` is mutually singular with respect to `μ`
and `s = t + μ.withDensityᵥ f`. In this case `t ... | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
| s | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [← toSignedMeasure_toJordanDecomposition s, JordanDecomposition.toSignedMeasure] | /-- **The Lebesgue Decomposition theorem between a signed measure and a measure**:
Given a signed measure `s` and a σ-finite measure `μ`, there exist a signed measure `t` and a
measurable and integrable function `f`, such that `t` is mutually singular with respect to `μ`
and `s = t + μ.withDensityᵥ f`. In this case `t ... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.205_0.HPGboz0rhL6sBes | /-- **The Lebesgue Decomposition theorem between a signed measure and a measure**:
Given a signed measure `s` and a σ-finite measure `μ`, there exist a signed measure `t` and a
measurable and integrable function `f`, such that `t` is mutually singular with respect to `μ`
and `s = t + μ.withDensityᵥ f`. In this case `t ... | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ singularPart s μ + withDensityᵥ μ (rnDeriv s μ) =
toSignedMeasure (toJordanDecomposition s).posPart - toSignedMeasure (toJordanDecomposition s).negPart | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [singularPart, rnDeriv,
withDensityᵥ_sub' (integrable_toReal_of_lintegral_ne_top _ _)
(integrable_toReal_of_lintegral_ne_top _ _),
withDensityᵥ_toReal, withDensityᵥ_toReal, sub_eq_add_neg, sub_eq_add_neg,
add_comm (s.toJordanDecomposition.posPart.singularPart μ).toSignedMeasure, ← add_assoc,
ad... | /-- **The Lebesgue Decomposition theorem between a signed measure and a measure**:
Given a signed measure `s` and a σ-finite measure `μ`, there exist a signed measure `t` and a
measurable and integrable function `f`, such that `t` is mutually singular with respect to `μ`
and `s = t + μ.withDensityᵥ f`. In this case `t ... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.205_0.HPGboz0rhL6sBes | /-- **The Lebesgue Decomposition theorem between a signed measure and a measure**:
Given a signed measure `s` and a σ-finite measure `μ`, there exist a signed measure `t` and a
measurable and integrable function `f`, such that `t` is mutually singular with respect to `μ`
and `s = t + μ.withDensityᵥ f`. In this case `t ... | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ toSignedMeasure
(Measure.singularPart (toJordanDecomposition s).posPart μ +
withDensity μ fun x => Measure.rnDeriv (toJordanDecomposition s).posPart μ x) -
toSignedMeasur... | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | convert rfl | /-- **The Lebesgue Decomposition theorem between a signed measure and a measure**:
Given a signed measure `s` and a σ-finite measure `μ`, there exist a signed measure `t` and a
measurable and integrable function `f`, such that `t` is mutually singular with respect to `μ`
and `s = t + μ.withDensityᵥ f`. In this case `t ... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.205_0.HPGboz0rhL6sBes | /-- **The Lebesgue Decomposition theorem between a signed measure and a measure**:
Given a signed measure `s` and a σ-finite measure `μ`, there exist a signed measure `t` and a
measurable and integrable function `f`, such that `t` is mutually singular with respect to `μ`
and `s = t + μ.withDensityᵥ f`. In this case `t ... | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case h.e'_3.h.e'_5.h.e'_3
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (toJordanDecomposition s).posPart =
Measure.singularPart (toJordanDecomposition s).posPart μ +
withDensity μ fun x => Measure.rnDeriv (toJordanDecomposition s... | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | exact s.toJordanDecomposition.posPart.haveLebesgueDecomposition_add μ | /-- **The Lebesgue Decomposition theorem between a signed measure and a measure**:
Given a signed measure `s` and a σ-finite measure `μ`, there exist a signed measure `t` and a
measurable and integrable function `f`, such that `t` is mutually singular with respect to `μ`
and `s = t + μ.withDensityᵥ f`. In this case `t ... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.205_0.HPGboz0rhL6sBes | /-- **The Lebesgue Decomposition theorem between a signed measure and a measure**:
Given a signed measure `s` and a σ-finite measure `μ`, there exist a signed measure `t` and a
measurable and integrable function `f`, such that `t` is mutually singular with respect to `μ`
and `s = t + μ.withDensityᵥ f`. In this case `t ... | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case h.e'_3.h.e'_6.h.e'_3
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (toJordanDecomposition s).negPart =
(withDensity μ fun x => Measure.rnDeriv (toJordanDecomposition s).negPart μ x) +
Measure.singularPart (toJordanDecompositi... | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [add_comm] | /-- **The Lebesgue Decomposition theorem between a signed measure and a measure**:
Given a signed measure `s` and a σ-finite measure `μ`, there exist a signed measure `t` and a
measurable and integrable function `f`, such that `t` is mutually singular with respect to `μ`
and `s = t + μ.withDensityᵥ f`. In this case `t ... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.205_0.HPGboz0rhL6sBes | /-- **The Lebesgue Decomposition theorem between a signed measure and a measure**:
Given a signed measure `s` and a σ-finite measure `μ`, there exist a signed measure `t` and a
measurable and integrable function `f`, such that `t` is mutually singular with respect to `μ`
and `s = t + μ.withDensityᵥ f`. In this case `t ... | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case h.e'_3.h.e'_6.h.e'_3
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ (toJordanDecomposition s).negPart =
Measure.singularPart (toJordanDecomposition s).negPart μ +
withDensity μ fun x => Measure.rnDeriv (toJordanDecomposition s... | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | exact s.toJordanDecomposition.negPart.haveLebesgueDecomposition_add μ | /-- **The Lebesgue Decomposition theorem between a signed measure and a measure**:
Given a signed measure `s` and a σ-finite measure `μ`, there exist a signed measure `t` and a
measurable and integrable function `f`, such that `t` is mutually singular with respect to `μ`
and `s = t + μ.withDensityᵥ f`. In this case `t ... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.205_0.HPGboz0rhL6sBes | /-- **The Lebesgue Decomposition theorem between a signed measure and a measure**:
Given a signed measure `s` and a σ-finite measure `μ`, there exist a signed measure `t` and a
measurable and integrable function `f`, such that `t` is mutually singular with respect to `μ`
and `s = t + μ.withDensityᵥ f`. In this case `t ... | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case hf
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ ∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ ≠ ⊤
case hf
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgu... | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | all_goals
first
| exact (lintegral_rnDeriv_lt_top _ _).ne
| measurability | /-- **The Lebesgue Decomposition theorem between a signed measure and a measure**:
Given a signed measure `s` and a σ-finite measure `μ`, there exist a signed measure `t` and a
measurable and integrable function `f`, such that `t` is mutually singular with respect to `μ`
and `s = t + μ.withDensityᵥ f`. In this case `t ... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.205_0.HPGboz0rhL6sBes | /-- **The Lebesgue Decomposition theorem between a signed measure and a measure**:
Given a signed measure `s` and a σ-finite measure `μ`, there exist a signed measure `t` and a
measurable and integrable function `f`, such that `t` is mutually singular with respect to `μ`
and `s = t + μ.withDensityᵥ f`. In this case `t ... | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case hf
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ ∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ ≠ ⊤ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | first
| exact (lintegral_rnDeriv_lt_top _ _).ne
| measurability | /-- **The Lebesgue Decomposition theorem between a signed measure and a measure**:
Given a signed measure `s` and a σ-finite measure `μ`, there exist a signed measure `t` and a
measurable and integrable function `f`, such that `t` is mutually singular with respect to `μ`
and `s = t + μ.withDensityᵥ f`. In this case `t ... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.205_0.HPGboz0rhL6sBes | /-- **The Lebesgue Decomposition theorem between a signed measure and a measure**:
Given a signed measure `s` and a σ-finite measure `μ`, there exist a signed measure `t` and a
measurable and integrable function `f`, such that `t` is mutually singular with respect to `μ`
and `s = t + μ.withDensityᵥ f`. In this case `t ... | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case hf
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ ∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ ≠ ⊤ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | exact (lintegral_rnDeriv_lt_top _ _).ne | /-- **The Lebesgue Decomposition theorem between a signed measure and a measure**:
Given a signed measure `s` and a σ-finite measure `μ`, there exist a signed measure `t` and a
measurable and integrable function `f`, such that `t` is mutually singular with respect to `μ`
and `s = t + μ.withDensityᵥ f`. In this case `t ... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.205_0.HPGboz0rhL6sBes | /-- **The Lebesgue Decomposition theorem between a signed measure and a measure**:
Given a signed measure `s` and a σ-finite measure `μ`, there exist a signed measure `t` and a
measurable and integrable function `f`, such that `t` is mutually singular with respect to `μ`
and `s = t + μ.withDensityᵥ f`. In this case `t ... | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
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