Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | rank int64 0 2.4k |
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import Mathlib.Topology.Algebra.GroupCompletion
import Mathlib.Topology.Algebra.InfiniteSum.Group
open UniformSpace.Completion
variable {Ξ± Ξ² : Type*} [AddCommGroup Ξ±] [UniformSpace Ξ±] [UniformAddGroup Ξ±]
theorem hasSum_iff_hasSum_compl (f : Ξ² β Ξ±) (a : Ξ±):
HasSum (toCompl β f) a β HasSum f a := (denseInducin... | Mathlib/Topology/Algebra/InfiniteSum/GroupCompletion.lean | 32 | 45 | theorem summable_iff_cauchySeq_finset_and_tsum_mem (f : Ξ² β Ξ±) :
Summable f β CauchySeq (fun s : Finset Ξ² β¦ β b in s, f b) β§
β' i, toCompl (f i) β Set.range toCompl := by |
classical
constructor
Β· rintro β¨a, haβ©
exact β¨ha.cauchySeq, ((summable_iff_summable_compl_and_tsum_mem f).mp β¨a, haβ©).2β©
Β· rintro β¨h_cauchy, h_tsumβ©
apply (summable_iff_summable_compl_and_tsum_mem f).mpr
constructor
Β· apply summable_iff_cauchySeq_finset.mpr
simp_rw [Function.comp_apply, β... | 1,375 |
import Mathlib.Topology.Algebra.InfiniteSum.Group
import Mathlib.Topology.Algebra.Star
noncomputable section
open Filter Finset Function
open scoped Topology
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*}
section ProdDomain
variable [CommMonoid Ξ±] [TopologicalSpace Ξ±]
@[to_additive]
| Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | 33 | 35 | theorem hasProd_pi_single [DecidableEq Ξ²] (b : Ξ²) (a : Ξ±) : HasProd (Pi.mulSingle b a) a := by |
convert hasProd_ite_eq b a
simp [Pi.mulSingle_apply]
| 1,376 |
import Mathlib.Topology.Algebra.InfiniteSum.Group
import Mathlib.Topology.Algebra.Star
noncomputable section
open Filter Finset Function
open scoped Topology
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*}
section ProdDomain
variable [CommMonoid Ξ±] [TopologicalSpace Ξ±]
@[to_additive]
theorem hasProd_pi_single [DecidableEq Ξ²] (... | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | 39 | 42 | theorem tprod_pi_single [DecidableEq Ξ²] (b : Ξ²) (a : Ξ±) : β' b', Pi.mulSingle b a b' = a := by |
rw [tprod_eq_mulSingle b]
Β· simp
Β· intro b' hb'; simp [hb']
| 1,376 |
import Mathlib.Topology.Algebra.InfiniteSum.Group
import Mathlib.Topology.Algebra.Star
noncomputable section
open Filter Finset Function
open scoped Topology
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*}
section ProdCodomain
variable [CommMonoid Ξ±] [TopologicalSpace Ξ±] [CommMonoid Ξ³] [TopologicalSpace Ξ³]
@[to_additive HasSum... | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | 68 | 70 | theorem HasProd.prod_mk {f : Ξ² β Ξ±} {g : Ξ² β Ξ³} {a : Ξ±} {b : Ξ³}
(hf : HasProd f a) (hg : HasProd g b) : HasProd (fun x β¦ (β¨f x, g xβ© : Ξ± Γ Ξ³)) β¨a, bβ© := by |
simp [HasProd, β prod_mk_prod, Filter.Tendsto.prod_mk_nhds hf hg]
| 1,376 |
import Mathlib.Topology.Algebra.InfiniteSum.Group
import Mathlib.Topology.Algebra.Star
noncomputable section
open Filter Finset Function
open scoped Topology
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*}
section ContinuousMul
variable [CommMonoid Ξ±] [TopologicalSpace Ξ±] [ContinuousMul Ξ±]
section RegularSpace
variable [Regul... | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | 84 | 101 | theorem HasProd.sigma {Ξ³ : Ξ² β Type*} {f : (Ξ£ b : Ξ², Ξ³ b) β Ξ±} {g : Ξ² β Ξ±} {a : Ξ±}
(ha : HasProd f a) (hf : β b, HasProd (fun c β¦ f β¨b, cβ©) (g b)) : HasProd g a := by |
classical
refine (atTop_basis.tendsto_iff (closed_nhds_basis a)).mpr ?_
rintro s β¨hs, hscβ©
rcases mem_atTop_sets.mp (ha hs) with β¨u, huβ©
use u.image Sigma.fst, trivial
intro bs hbs
simp only [Set.mem_preimage, ge_iff_le, Finset.le_iff_subset] at hu
have : Tendsto (fun t : Finset (Ξ£b, Ξ³ b) β¦ β p β t.fil... | 1,376 |
import Mathlib.Topology.Algebra.InfiniteSum.Constructions
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.infinite_sum.module from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*}
open Filter Finset Function
variable {ΞΉ ΞΊ R Rβ M Mβ... | Mathlib/Topology/Algebra/InfiniteSum/Module.lean | 167 | 178 | theorem ContinuousLinearEquiv.tsum_eq_iff [T2Space M] [T2Space Mβ] {f : ΞΉ β M} (e : M βSL[Ο] Mβ)
{y : Mβ} : (β' z, e (f z)) = y β β' z, f z = e.symm y := by |
by_cases hf : Summable f
Β· exact
β¨fun h β¦ (e.hasSum.mp ((e.summable.mpr hf).hasSum_iff.mpr h)).tsum_eq, fun h β¦
(e.hasSum.mpr (hf.hasSum_iff.mpr h)).tsum_eqβ©
Β· have hf' : Β¬Summable fun z β¦ e (f z) := fun h β¦ hf (e.summable.mp h)
rw [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable ... | 1,377 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Topology.Algebra.InfiniteSum.Constructions
import Mathlib.Topology.Algebra.Ring.Basic
#align_import topology.algebra.infinite_sum.ring from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Filter Finset Function
open... | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | 34 | 35 | theorem HasSum.mul_left (aβ) (h : HasSum f aβ) : HasSum (fun i β¦ aβ * f i) (aβ * aβ) := by |
simpa only using h.map (AddMonoidHom.mulLeft aβ) (continuous_const.mul continuous_id)
| 1,378 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Topology.Algebra.InfiniteSum.Constructions
import Mathlib.Topology.Algebra.Ring.Basic
#align_import topology.algebra.infinite_sum.ring from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Filter Finset Function
open... | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | 38 | 39 | theorem HasSum.mul_right (aβ) (hf : HasSum f aβ) : HasSum (fun i β¦ f i * aβ) (aβ * aβ) := by |
simpa only using hf.map (AddMonoidHom.mulRight aβ) (continuous_id.mul continuous_const)
| 1,378 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Topology.Algebra.InfiniteSum.Constructions
import Mathlib.Topology.Algebra.Ring.Basic
#align_import topology.algebra.infinite_sum.ring from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Filter Finset Function
open... | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | 208 | 213 | theorem summable_sum_mul_antidiagonal_of_summable_mul
(h : Summable fun x : A Γ A β¦ f x.1 * g x.2) :
Summable fun n β¦ β kl β antidiagonal n, f kl.1 * g kl.2 := by |
rw [summable_mul_prod_iff_summable_mul_sigma_antidiagonal] at h
conv => congr; ext; rw [β Finset.sum_finset_coe, β tsum_fintype]
exact h.sigma' fun n β¦ (hasSum_fintype _).summable
| 1,378 |
import Mathlib.Topology.Algebra.InfiniteSum.Group
import Mathlib.Logic.Encodable.Lattice
noncomputable section
open Filter Finset Function Encodable
open scoped Topology
variable {M : Type*} [CommMonoid M] [TopologicalSpace M] {m m' : M}
variable {G : Type*} [CommGroup G] {g g' : G}
-- don't declare [Topologic... | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | 62 | 65 | theorem prod_range_mul {f : β β M} {k : β} (h : HasProd (fun n β¦ f (n + k)) m) :
HasProd f ((β i β range k, f i) * m) := by |
refine ((range k).hasProd f).mul_compl ?_
rwa [β (notMemRangeEquiv k).symm.hasProd_iff]
| 1,379 |
import Mathlib.Topology.Algebra.InfiniteSum.Group
import Mathlib.Logic.Encodable.Lattice
noncomputable section
open Filter Finset Function Encodable
open scoped Topology
variable {M : Type*} [CommMonoid M] [TopologicalSpace M] {m m' : M}
variable {G : Type*} [CommGroup G] {g g' : G}
-- don't declare [Topologic... | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | 68 | 70 | theorem zero_mul {f : β β M} (h : HasProd (fun n β¦ f (n + 1)) m) :
HasProd f (f 0 * m) := by |
simpa only [prod_range_one] using h.prod_range_mul
| 1,379 |
import Mathlib.Topology.Algebra.InfiniteSum.Group
import Mathlib.Logic.Encodable.Lattice
noncomputable section
open Filter Finset Function Encodable
open scoped Topology
variable {M : Type*} [CommMonoid M] [TopologicalSpace M] {m m' : M}
variable {G : Type*} [CommGroup G] {g g' : G}
-- don't declare [Topologic... | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | 73 | 78 | theorem even_mul_odd {f : β β M} (he : HasProd (fun k β¦ f (2 * k)) m)
(ho : HasProd (fun k β¦ f (2 * k + 1)) m') : HasProd f (m * m') := by |
have := mul_right_injectiveβ (two_ne_zero' β)
replace ho := ((add_left_injective 1).comp this).hasProd_range_iff.2 ho
refine (this.hasProd_range_iff.2 he).mul_isCompl ?_ ho
simpa [(Β· β Β·)] using Nat.isCompl_even_odd
| 1,379 |
import Mathlib.Topology.Algebra.InfiniteSum.Group
import Mathlib.Logic.Encodable.Lattice
noncomputable section
open Filter Finset Function Encodable
open scoped Topology
variable {M : Type*} [CommMonoid M] [TopologicalSpace M] {m m' : M}
variable {G : Type*} [CommGroup G] {g g' : G}
-- don't declare [Topologic... | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | 88 | 92 | theorem hasProd_iff_tendsto_nat [T2Space M] {f : β β M} (hf : Multipliable f) :
HasProd f m β Tendsto (fun n : β β¦ β i β range n, f i) atTop (π m) := by |
refine β¨fun h β¦ h.tendsto_prod_nat, fun h β¦ ?_β©
rw [tendsto_nhds_unique h hf.hasProd.tendsto_prod_nat]
exact hf.hasProd
| 1,379 |
import Mathlib.Topology.Algebra.InfiniteSum.Group
import Mathlib.Logic.Encodable.Lattice
noncomputable section
open Filter Finset Function Encodable
open scoped Topology
variable {M : Type*} [CommMonoid M] [TopologicalSpace M] {m m' : M}
variable {G : Type*} [CommGroup G] {g g' : G}
-- don't declare [Topologic... | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | 124 | 132 | theorem tprod_iSup_decodeβ [CompleteLattice Ξ±] (m : Ξ± β M) (m0 : m β₯ = 1) (s : Ξ² β Ξ±) :
β' i : β, m (β¨ b β decodeβ Ξ² i, s b) = β' b : Ξ², m (s b) := by |
rw [β tprod_extend_one (@encode_injective Ξ² _)]
refine tprod_congr fun n β¦ ?_
rcases em (n β Set.range (encode : Ξ² β β)) with β¨a, rflβ© | hn
Β· simp [encode_injective.extend_apply]
Β· rw [extend_apply' _ _ _ hn]
rw [β decodeβ_ne_none_iff, ne_eq, not_not] at hn
simp [hn, m0]
| 1,379 |
import Mathlib.Topology.Algebra.InfiniteSum.Group
import Mathlib.Logic.Encodable.Lattice
noncomputable section
open Filter Finset Function Encodable
open scoped Topology
variable {M : Type*} [CommMonoid M] [TopologicalSpace M] {m m' : M}
variable {G : Type*} [CommGroup G] {g g' : G}
-- don't declare [Topologic... | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | 218 | 221 | theorem hasProd_nat_add_iff {f : β β G} (k : β) :
HasProd (fun n β¦ f (n + k)) g β HasProd f (g * β i β range k, f i) := by |
refine Iff.trans ?_ (range k).hasProd_compl_iff
rw [β (notMemRangeEquiv k).symm.hasProd_iff, Function.comp_def, coe_notMemRangeEquiv_symm]
| 1,379 |
import Mathlib.Topology.Algebra.InfiniteSum.Group
import Mathlib.Logic.Encodable.Lattice
noncomputable section
open Filter Finset Function Encodable
open scoped Topology
variable {M : Type*} [CommMonoid M] [TopologicalSpace M] {m m' : M}
variable {G : Type*} [CommGroup G] {g g' : G}
-- don't declare [Topologic... | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | 273 | 285 | theorem cauchySeq_finset_iff_nat_tprod_vanishing {f : β β G} :
(CauchySeq fun s : Finset β β¦ β n β s, f n) β
β e β π (1 : G), β N : β, β t β {n | N β€ n}, (β' n : t, f n) β e := by |
refine cauchySeq_finset_iff_tprod_vanishing.trans β¨fun vanish e he β¦ ?_, fun vanish e he β¦ ?_β©
Β· obtain β¨s, hsβ© := vanish e he
refine β¨if h : s.Nonempty then s.max' h + 1 else 0,
fun t ht β¦ hs _ <| Set.disjoint_left.mpr ?_β©
split_ifs at ht with h
Β· exact fun m hmt hms β¦ (s.le_max' _ hms).not_lt (... | 1,379 |
import Mathlib.Topology.Algebra.InfiniteSum.Group
import Mathlib.Logic.Encodable.Lattice
noncomputable section
open Filter Finset Function Encodable
open scoped Topology
variable {M : Type*} [CommMonoid M] [TopologicalSpace M] {m m' : M}
variable {G : Type*} [CommGroup G] {g g' : G}
-- don't declare [Topologic... | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | 290 | 292 | theorem multipliable_iff_nat_tprod_vanishing {f : β β G} : Multipliable f β
β e β π 1, β N : β, β t β {n | N β€ n}, (β' n : t, f n) β e := by |
rw [multipliable_iff_cauchySeq_finset, cauchySeq_finset_iff_nat_tprod_vanishing]
| 1,379 |
import Mathlib.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Topology.Order.Basic
#align_import analysis.convex.strict from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
open Set
open Convex Pointwise
variable {π π E F Ξ² : Type*}
open Function Se... | Mathlib/Analysis/Convex/Strict.lean | 67 | 70 | theorem strictConvex_univ : StrictConvex π (univ : Set E) := by |
intro x _ y _ _ a b _ _ _
rw [interior_univ]
exact mem_univ _
| 1,380 |
import Mathlib.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Topology.Order.Basic
#align_import analysis.convex.strict from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
open Set
open Convex Pointwise
variable {π π E F Ξ² : Type*}
open Function Se... | Mathlib/Analysis/Convex/Strict.lean | 85 | 92 | theorem Directed.strictConvex_iUnion {ΞΉ : Sort*} {s : ΞΉ β Set E} (hdir : Directed (Β· β Β·) s)
(hs : β β¦i : ΞΉβ¦, StrictConvex π (s i)) : StrictConvex π (β i, s i) := by |
rintro x hx y hy hxy a b ha hb hab
rw [mem_iUnion] at hx hy
obtain β¨i, hxβ© := hx
obtain β¨j, hyβ© := hy
obtain β¨k, hik, hjkβ© := hdir i j
exact interior_mono (subset_iUnion s k) (hs (hik hx) (hjk hy) hxy ha hb hab)
| 1,380 |
import Mathlib.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Topology.Order.Basic
#align_import analysis.convex.strict from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
open Set
open Convex Pointwise
variable {π π E F Ξ² : Type*}
open Function Se... | Mathlib/Analysis/Convex/Strict.lean | 95 | 98 | theorem DirectedOn.strictConvex_sUnion {S : Set (Set E)} (hdir : DirectedOn (Β· β Β·) S)
(hS : β s β S, StrictConvex π s) : StrictConvex π (ββ S) := by |
rw [sUnion_eq_iUnion]
exact (directedOn_iff_directed.1 hdir).strictConvex_iUnion fun s => hS _ s.2
| 1,380 |
import Mathlib.Topology.Baire.Lemmas
import Mathlib.Topology.Algebra.Group.Basic
open scoped Topology Pointwise
open MulAction Set Function
variable {G X : Type*} [TopologicalSpace G] [TopologicalSpace X]
[Group G] [TopologicalGroup G] [MulAction G X]
[SigmaCompactSpace G] [BaireSpace X] [T2Space X]
[Contin... | Mathlib/Topology/Algebra/Group/OpenMapping.lean | 37 | 88 | theorem smul_singleton_mem_nhds_of_sigmaCompact
{U : Set G} (hU : U β π 1) (x : X) : U β’ {x} β π x := by |
/- Consider a small closed neighborhood `V` of the identity. Then the group is covered by
countably many translates of `V`, say `gα΅’ V`. Let also `Kβ` be a sequence of compact sets covering
the space. Then the image of `Kβ β© gα΅’ V` in the orbit is compact, and their unions covers the
space. By Baire, one of them... | 1,381 |
import Mathlib.Topology.Baire.Lemmas
import Mathlib.Topology.Algebra.Group.Basic
open scoped Topology Pointwise
open MulAction Set Function
variable {G X : Type*} [TopologicalSpace G] [TopologicalSpace X]
[Group G] [TopologicalGroup G] [MulAction G X]
[SigmaCompactSpace G] [BaireSpace X] [T2Space X]
[Contin... | Mathlib/Topology/Algebra/Group/OpenMapping.lean | 96 | 107 | theorem isOpenMap_smul_of_sigmaCompact (x : X) : IsOpenMap (fun (g : G) β¦ g β’ x) := by |
/- We have already proved the theorem around the basepoint of the orbit, in
`smul_singleton_mem_nhds_of_sigmaCompact`. The general statement follows around an arbitrary
point by changing basepoints. -/
simp_rw [isOpenMap_iff_nhds_le, Filter.le_map_iff]
intro g U hU
have : (Β· β’ x) = (Β· β’ (g β’ x)) β (Β· * gβ»ΒΉ... | 1,381 |
import Mathlib.Topology.Baire.Lemmas
import Mathlib.Topology.Algebra.Group.Basic
open scoped Topology Pointwise
open MulAction Set Function
variable {G X : Type*} [TopologicalSpace G] [TopologicalSpace X]
[Group G] [TopologicalGroup G] [MulAction G X]
[SigmaCompactSpace G] [BaireSpace X] [T2Space X]
[Contin... | Mathlib/Topology/Algebra/Group/OpenMapping.lean | 112 | 121 | theorem MonoidHom.isOpenMap_of_sigmaCompact
{H : Type*} [Group H] [TopologicalSpace H] [BaireSpace H] [T2Space H] [ContinuousMul H]
(f : G β* H) (hf : Function.Surjective f) (h'f : Continuous f) :
IsOpenMap f := by |
let A : MulAction G H := MulAction.compHom _ f
have : ContinuousSMul G H := continuousSMul_compHom h'f
have : IsPretransitive G H := isPretransitive_compHom hf
have : f = (fun (g : G) β¦ g β’ (1 : H)) := by simp [MulAction.compHom_smul_def]
rw [this]
exact isOpenMap_smul_of_sigmaCompact _
| 1,381 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.Group.Indicator
import Mathlib.Order.LiminfLimsup
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Data.Set.La... | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | 339 | 385 | theorem Antitone.map_limsSup_of_continuousAt {F : Filter R} [NeBot F] {f : R β S}
(f_decr : Antitone f) (f_cont : ContinuousAt f F.limsSup)
(bdd_above : F.IsBounded (Β· β€ Β·) := by | isBoundedDefault)
(bdd_below : F.IsBounded (Β· β₯ Β·) := by isBoundedDefault) :
f F.limsSup = F.liminf f := by
have cobdd : F.IsCobounded (Β· β€ Β·) := bdd_below.isCobounded_flip
apply le_antisymm
Β· rw [limsSup, f_decr.map_sInf_of_continuousAt' f_cont bdd_above cobdd]
apply le_of_forall_lt
intro c hc
... | 1,382 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.Group.Indicator
import Mathlib.Order.LiminfLimsup
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Data.Set.La... | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | 476 | 479 | theorem iInf_eq_of_forall_le_of_tendsto {x : R} {as : ΞΉ β R} (x_le : β i, x β€ as i) {F : Filter ΞΉ}
[Filter.NeBot F] (as_lim : Filter.Tendsto as F (π x)) : β¨
i, as i = x := by |
refine iInf_eq_of_forall_ge_of_forall_gt_exists_lt (fun i β¦ x_le i) ?_
apply fun w x_lt_w β¦ βΉFilter.NeBot FβΊ.nonempty_of_mem (eventually_lt_of_tendsto_lt x_lt_w as_lim)
| 1,382 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.Group.Indicator
import Mathlib.Order.LiminfLimsup
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Data.Set.La... | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | 487 | 498 | theorem iUnion_Ici_eq_Ioi_of_lt_of_tendsto (x : R) {as : ΞΉ β R} (x_lt : β i, x < as i)
{F : Filter ΞΉ} [Filter.NeBot F] (as_lim : Filter.Tendsto as F (π x)) :
β i : ΞΉ, Ici (as i) = Ioi x := by |
have obs : x β range as := by
intro maybe_x_is
rcases mem_range.mp maybe_x_is with β¨i, hiβ©
simpa only [hi, lt_self_iff_false] using x_lt i
-- Porting note: `rw at *` was too destructive. Let's only rewrite `obs` and the goal.
have := iInf_eq_of_forall_le_of_tendsto (fun i β¦ (x_lt i).le) as_lim
rw [... | 1,382 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.Group.Indicator
import Mathlib.Order.LiminfLimsup
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Data.Set.La... | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | 511 | 565 | theorem limsup_eq_tendsto_sum_indicator_nat_atTop (s : β β Set Ξ±) :
limsup s atTop = { Ο | Tendsto
(fun n β¦ β k β Finset.range n, (s (k + 1)).indicator (1 : Ξ± β β) Ο) atTop atTop } := by |
ext Ο
simp only [limsup_eq_iInf_iSup_of_nat, ge_iff_le, Set.iSup_eq_iUnion, Set.iInf_eq_iInter,
Set.mem_iInter, Set.mem_iUnion, exists_prop]
constructor
Β· intro hΟ
refine tendsto_atTop_atTop_of_monotone' (fun n m hnm β¦ Finset.sum_mono_set_of_nonneg
(fun i β¦ Set.indicator_nonneg (fun _ _ β¦ zero_le... | 1,382 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.Group.Indicator
import Mathlib.Order.LiminfLimsup
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Data.Set.La... | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | 568 | 578 | theorem limsup_eq_tendsto_sum_indicator_atTop (R : Type*) [StrictOrderedSemiring R] [Archimedean R]
(s : β β Set Ξ±) : limsup s atTop = { Ο | Tendsto
(fun n β¦ β k β Finset.range n, (s (k + 1)).indicator (1 : Ξ± β R) Ο) atTop atTop } := by |
rw [limsup_eq_tendsto_sum_indicator_nat_atTop s]
ext Ο
simp only [Set.mem_setOf_eq]
rw [(_ : (fun n β¦ β k β Finset.range n, (s (k + 1)).indicator (1 : Ξ± β R) Ο) = fun n β¦
β(β k β Finset.range n, (s (k + 1)).indicator (1 : Ξ± β β) Ο))]
Β· exact tendsto_natCast_atTop_iff.symm
Β· ext n
simp only [Set.ind... | 1,382 |
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import measure_theory.measure.measure_space from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 107 | 109 | theorem ae_uIoc_iff [LinearOrder Ξ±] {a b : Ξ±} {P : Ξ± β Prop} :
(βα΅ x βΞΌ, x β Ξ a b β P x) β (βα΅ x βΞΌ, x β Ioc a b β P x) β§ βα΅ x βΞΌ, x β Ioc b a β P x := by |
simp only [uIoc_eq_union, mem_union, or_imp, eventually_and]
| 1,383 |
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import measure_theory.measure.measure_space from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 128 | 132 | theorem measure_union_add_inter (s : Set Ξ±) (ht : MeasurableSet t) :
ΞΌ (s βͺ t) + ΞΌ (s β© t) = ΞΌ s + ΞΌ t := by |
rw [β measure_inter_add_diff (s βͺ t) ht, Set.union_inter_cancel_right, union_diff_right, β
measure_inter_add_diff s ht]
ac_rfl
| 1,383 |
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import measure_theory.measure.measure_space from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 135 | 137 | theorem measure_union_add_inter' (hs : MeasurableSet s) (t : Set Ξ±) :
ΞΌ (s βͺ t) + ΞΌ (s β© t) = ΞΌ s + ΞΌ t := by |
rw [union_comm, inter_comm, measure_union_add_inter t hs, add_comm]
| 1,383 |
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import measure_theory.measure.measure_space from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 152 | 157 | theorem measure_biUnionβ {s : Set Ξ²} {f : Ξ² β Set Ξ±} (hs : s.Countable)
(hd : s.Pairwise (AEDisjoint ΞΌ on f)) (h : β b β s, NullMeasurableSet (f b) ΞΌ) :
ΞΌ (β b β s, f b) = β' p : s, ΞΌ (f p) := by |
haveI := hs.toEncodable
rw [biUnion_eq_iUnion]
exact measure_iUnionβ (hd.on_injective Subtype.coe_injective fun x => x.2) fun x => h x x.2
| 1,383 |
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import measure_theory.measure.measure_space from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 165 | 167 | theorem measure_sUnionβ {S : Set (Set Ξ±)} (hs : S.Countable) (hd : S.Pairwise (AEDisjoint ΞΌ))
(h : β s β S, NullMeasurableSet s ΞΌ) : ΞΌ (ββ S) = β' s : S, ΞΌ s := by |
rw [sUnion_eq_biUnion, measure_biUnionβ hs hd h]
| 1,383 |
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import measure_theory.measure.measure_space from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 170 | 172 | theorem measure_sUnion {S : Set (Set Ξ±)} (hs : S.Countable) (hd : S.Pairwise Disjoint)
(h : β s β S, MeasurableSet s) : ΞΌ (ββ S) = β' s : S, ΞΌ s := by |
rw [sUnion_eq_biUnion, measure_biUnion hs hd h]
| 1,383 |
import Mathlib.MeasureTheory.Measure.MeasureSpace
#align_import measure_theory.covering.vitali_family from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open MeasureTheory Metric Set Filter TopologicalSpace MeasureTheory.Measure
open Filter MeasureTheory Topology
variable {Ξ± : Type*}... | Mathlib/MeasureTheory/Covering/VitaliFamily.lean | 226 | 228 | theorem _root_.Filter.HasBasis.vitaliFamily {ΞΉ : Sort*} {p : ΞΉ β Prop} {s : ΞΉ β Set Ξ±} {x : Ξ±}
(h : (π x).HasBasis p s) : (v.filterAt x).HasBasis p (fun i β¦ {t β v.setsAt x | t β s i}) := by |
simpa only [β Set.setOf_inter_eq_sep] using h.smallSets.inf_principal _
| 1,384 |
import Mathlib.MeasureTheory.Measure.MeasureSpace
#align_import measure_theory.covering.vitali_family from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open MeasureTheory Metric Set Filter TopologicalSpace MeasureTheory.Measure
open Filter MeasureTheory Topology
variable {Ξ± : Type*}... | Mathlib/MeasureTheory/Covering/VitaliFamily.lean | 234 | 236 | theorem mem_filterAt_iff {x : Ξ±} {s : Set (Set Ξ±)} :
s β v.filterAt x β β Ξ΅ > (0 : β), β a β v.setsAt x, a β closedBall x Ξ΅ β a β s := by |
simp only [(v.filterAt_basis_closedBall x).mem_iff, β and_imp, subset_def, mem_setOf]
| 1,384 |
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Measure.MeasureSpace
namespace MeasureTheory
namespace Measure
variable {M : Type*} [Monoid M] [MeasurableSpace M]
@[to_additive conv "Additive convolution of measures."]
noncomputable def mconv (ΞΌ : Measure M) (Ξ½ : Measure M) :
... | Mathlib/MeasureTheory/Group/Convolution.lean | 41 | 46 | theorem dirac_one_mconv [MeasurableMulβ M] (ΞΌ : Measure M) [SFinite ΞΌ] :
(Measure.dirac 1) β ΞΌ = ΞΌ := by |
unfold mconv
rw [MeasureTheory.Measure.dirac_prod, map_map]
Β· simp only [Function.comp_def, one_mul, map_id']
all_goals { measurability }
| 1,385 |
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Measure.MeasureSpace
namespace MeasureTheory
namespace Measure
variable {M : Type*} [Monoid M] [MeasurableSpace M]
@[to_additive conv "Additive convolution of measures."]
noncomputable def mconv (ΞΌ : Measure M) (Ξ½ : Measure M) :
... | Mathlib/MeasureTheory/Group/Convolution.lean | 50 | 55 | theorem mconv_dirac_one [MeasurableMulβ M]
(ΞΌ : Measure M) [SFinite ΞΌ] : ΞΌ β (Measure.dirac 1) = ΞΌ := by |
unfold mconv
rw [MeasureTheory.Measure.prod_dirac, map_map]
Β· simp only [Function.comp_def, mul_one, map_id']
all_goals { measurability }
| 1,385 |
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Measure.MeasureSpace
namespace MeasureTheory
namespace Measure
variable {M : Type*} [Monoid M] [MeasurableSpace M]
@[to_additive conv "Additive convolution of measures."]
noncomputable def mconv (ΞΌ : Measure M) (Ξ½ : Measure M) :
... | Mathlib/MeasureTheory/Group/Convolution.lean | 59 | 61 | theorem mconv_zero (ΞΌ : Measure M) : (0 : Measure M) β ΞΌ = (0 : Measure M) := by |
unfold mconv
simp
| 1,385 |
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Measure.MeasureSpace
namespace MeasureTheory
namespace Measure
variable {M : Type*} [Monoid M] [MeasurableSpace M]
@[to_additive conv "Additive convolution of measures."]
noncomputable def mconv (ΞΌ : Measure M) (Ξ½ : Measure M) :
... | Mathlib/MeasureTheory/Group/Convolution.lean | 65 | 67 | theorem zero_mconv (ΞΌ : Measure M) : ΞΌ β (0 : Measure M) = (0 : Measure M) := by |
unfold mconv
simp
| 1,385 |
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Measure.MeasureSpace
namespace MeasureTheory
namespace Measure
variable {M : Type*} [Monoid M] [MeasurableSpace M]
@[to_additive conv "Additive convolution of measures."]
noncomputable def mconv (ΞΌ : Measure M) (Ξ½ : Measure M) :
... | Mathlib/MeasureTheory/Group/Convolution.lean | 70 | 74 | theorem mconv_add [MeasurableMulβ M] (ΞΌ : Measure M) (Ξ½ : Measure M) (Ο : Measure M) [SFinite ΞΌ]
[SFinite Ξ½] [SFinite Ο] : ΞΌ β (Ξ½ + Ο) = ΞΌ β Ξ½ + ΞΌ β Ο := by |
unfold mconv
rw [prod_add, map_add]
measurability
| 1,385 |
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Measure.MeasureSpace
namespace MeasureTheory
namespace Measure
variable {M : Type*} [Monoid M] [MeasurableSpace M]
@[to_additive conv "Additive convolution of measures."]
noncomputable def mconv (ΞΌ : Measure M) (Ξ½ : Measure M) :
... | Mathlib/MeasureTheory/Group/Convolution.lean | 77 | 81 | theorem add_mconv [MeasurableMulβ M] (ΞΌ : Measure M) (Ξ½ : Measure M) (Ο : Measure M) [SFinite ΞΌ]
[SFinite Ξ½] [SFinite Ο] : (ΞΌ + Ξ½) β Ο = ΞΌ β Ο + Ξ½ β Ο := by |
unfold mconv
rw [add_prod, map_add]
measurability
| 1,385 |
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Measure.MeasureSpace
namespace MeasureTheory
namespace Measure
variable {M : Type*} [Monoid M] [MeasurableSpace M]
@[to_additive conv "Additive convolution of measures."]
noncomputable def mconv (ΞΌ : Measure M) (Ξ½ : Measure M) :
... | Mathlib/MeasureTheory/Group/Convolution.lean | 85 | 90 | theorem mconv_comm {M : Type*} [CommMonoid M] [MeasurableSpace M] [MeasurableMulβ M] (ΞΌ : Measure M)
(Ξ½ : Measure M) [SFinite ΞΌ] [SFinite Ξ½] : ΞΌ β Ξ½ = Ξ½ β ΞΌ := by |
unfold mconv
rw [β prod_swap, map_map]
Β· simp [Function.comp_def, mul_comm]
all_goals { measurability }
| 1,385 |
import Mathlib.MeasureTheory.Measure.MeasureSpace
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter MeasurableSpace ENNReal Function
variable {R Ξ± Ξ² Ξ΄ Ξ³ ΞΉ : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace Ξ±} [MeasurableSpace Ξ²] [MeasurableSpace Ξ³]
variable {ΞΌ ΞΌβ ΞΌβ ΞΌβ Ξ½ Ξ½' Ξ½... | Mathlib/MeasureTheory/Measure/Restrict.lean | 56 | 59 | theorem restrict_toOuterMeasure_eq_toOuterMeasure_restrict (h : MeasurableSet s) :
(ΞΌ.restrict s).toOuterMeasure = OuterMeasure.restrict s ΞΌ.toOuterMeasure := by |
simp_rw [restrict, restrictβ, liftLinear, LinearMap.coe_mk, AddHom.coe_mk,
toMeasure_toOuterMeasure, OuterMeasure.restrict_trim h, ΞΌ.trimmed]
| 1,386 |
import Mathlib.MeasureTheory.Measure.MeasureSpace
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter MeasurableSpace ENNReal Function
variable {R Ξ± Ξ² Ξ΄ Ξ³ ΞΉ : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace Ξ±} [MeasurableSpace Ξ²] [MeasurableSpace Ξ³]
variable {ΞΌ ΞΌβ ΞΌβ ΞΌβ Ξ½ Ξ½' Ξ½... | Mathlib/MeasureTheory/Measure/Restrict.lean | 62 | 64 | theorem restrict_applyβ (ht : NullMeasurableSet t (ΞΌ.restrict s)) : ΞΌ.restrict s t = ΞΌ (t β© s) := by |
rw [β restrictβ_apply, restrictβ, liftLinear_applyβ _ ht, OuterMeasure.restrict_apply,
coe_toOuterMeasure]
| 1,386 |
import Mathlib.MeasureTheory.Measure.MeasureSpace
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter MeasurableSpace ENNReal Function
variable {R Ξ± Ξ² Ξ΄ Ξ³ ΞΉ : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace Ξ±} [MeasurableSpace Ξ²] [MeasurableSpace Ξ³]
variable {ΞΌ ΞΌβ ΞΌβ ΞΌβ Ξ½ Ξ½' Ξ½... | Mathlib/MeasureTheory/Measure/Restrict.lean | 104 | 107 | theorem restrict_apply' (hs : MeasurableSet s) : ΞΌ.restrict s t = ΞΌ (t β© s) := by |
rw [β toOuterMeasure_apply,
Measure.restrict_toOuterMeasure_eq_toOuterMeasure_restrict hs,
OuterMeasure.restrict_apply s t _, toOuterMeasure_apply]
| 1,386 |
import Mathlib.MeasureTheory.Measure.MeasureSpace
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter MeasurableSpace ENNReal Function
variable {R Ξ± Ξ² Ξ΄ Ξ³ ΞΉ : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace Ξ±} [MeasurableSpace Ξ²] [MeasurableSpace Ξ³]
variable {ΞΌ ΞΌβ ΞΌβ ΞΌβ Ξ½ Ξ½' Ξ½... | Mathlib/MeasureTheory/Measure/Restrict.lean | 110 | 113 | theorem restrict_applyβ' (hs : NullMeasurableSet s ΞΌ) : ΞΌ.restrict s t = ΞΌ (t β© s) := by |
rw [β restrict_congr_set hs.toMeasurable_ae_eq,
restrict_apply' (measurableSet_toMeasurable _ _),
measure_congr ((ae_eq_refl t).inter hs.toMeasurable_ae_eq)]
| 1,386 |
import Mathlib.MeasureTheory.Measure.MeasureSpace
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter MeasurableSpace ENNReal Function
variable {R Ξ± Ξ² Ξ΄ Ξ³ ΞΉ : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace Ξ±} [MeasurableSpace Ξ²] [MeasurableSpace Ξ³]
variable {ΞΌ ΞΌβ ΞΌβ ΞΌβ Ξ½ Ξ½' Ξ½... | Mathlib/MeasureTheory/Measure/Restrict.lean | 124 | 130 | theorem restrict_eq_self (h : s β t) : ΞΌ.restrict t s = ΞΌ s :=
(le_iff'.1 restrict_le_self s).antisymm <|
calc
ΞΌ s β€ ΞΌ (toMeasurable (ΞΌ.restrict t) s β© t) :=
measure_mono (subset_inter (subset_toMeasurable _ _) h)
_ = ΞΌ.restrict t s := by |
rw [β restrict_apply (measurableSet_toMeasurable _ _), measure_toMeasurable]
| 1,386 |
import Mathlib.MeasureTheory.Measure.MeasureSpace
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter MeasurableSpace ENNReal Function
variable {R Ξ± Ξ² Ξ΄ Ξ³ ΞΉ : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace Ξ±} [MeasurableSpace Ξ²] [MeasurableSpace Ξ³]
variable {ΞΌ ΞΌβ ΞΌβ ΞΌβ Ξ½ Ξ½' Ξ½... | Mathlib/MeasureTheory/Measure/Restrict.lean | 140 | 141 | theorem restrict_apply_univ (s : Set Ξ±) : ΞΌ.restrict s univ = ΞΌ s := by |
rw [restrict_apply MeasurableSet.univ, Set.univ_inter]
| 1,386 |
import Mathlib.MeasureTheory.Measure.Restrict
#align_import measure_theory.measure.mutually_singular from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
open Set
open MeasureTheory NNReal ENNReal
namespace MeasureTheory
namespace Measure
variable {Ξ± : Type*} {m0 : MeasurableSpace Ξ±}... | Mathlib/MeasureTheory/Measure/MutuallySingular.lean | 48 | 52 | theorem mk {s t : Set Ξ±} (hs : ΞΌ s = 0) (ht : Ξ½ t = 0) (hst : univ β s βͺ t) :
MutuallySingular ΞΌ Ξ½ := by |
use toMeasurable ΞΌ s, measurableSet_toMeasurable _ _, (measure_toMeasurable _).trans hs
refine measure_mono_null (fun x hx => (hst trivial).resolve_left fun hxs => hx ?_) ht
exact subset_toMeasurable _ _ hxs
| 1,387 |
import Mathlib.MeasureTheory.Measure.Restrict
#align_import measure_theory.measure.mutually_singular from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
open Set
open MeasureTheory NNReal ENNReal
namespace MeasureTheory
namespace Measure
variable {Ξ± : Type*} {m0 : MeasurableSpace Ξ±}... | Mathlib/MeasureTheory/Measure/MutuallySingular.lean | 114 | 120 | theorem sum_left {ΞΉ : Type*} [Countable ΞΉ] {ΞΌ : ΞΉ β Measure Ξ±} : sum ΞΌ ββ Ξ½ β β i, ΞΌ i ββ Ξ½ := by |
refine β¨fun h i => h.mono (le_sum _ _) le_rfl, fun H => ?_β©
choose s hsm hsΞΌ hsΞ½ using H
refine β¨β i, s i, MeasurableSet.iInter hsm, ?_, ?_β©
Β· rw [sum_apply _ (MeasurableSet.iInter hsm), ENNReal.tsum_eq_zero]
exact fun i => measure_mono_null (iInter_subset _ _) (hsΞΌ i)
Β· rwa [compl_iInter, measure_iUnion... | 1,387 |
import Mathlib.MeasureTheory.Measure.Restrict
#align_import measure_theory.measure.mutually_singular from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
open Set
open MeasureTheory NNReal ENNReal
namespace MeasureTheory
namespace Measure
variable {Ξ± : Type*} {m0 : MeasurableSpace Ξ±}... | Mathlib/MeasureTheory/Measure/MutuallySingular.lean | 129 | 130 | theorem add_left_iff : ΞΌβ + ΞΌβ ββ Ξ½ β ΞΌβ ββ Ξ½ β§ ΞΌβ ββ Ξ½ := by |
rw [β sum_cond, sum_left, Bool.forall_bool, cond, cond, and_comm]
| 1,387 |
import Mathlib.MeasureTheory.Measure.Restrict
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter Function MeasurableSpace ENNReal
variable {Ξ± Ξ² Ξ΄ ΞΉ : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace Ξ±} [MeasurableSpace Ξ²] {ΞΌ Ξ½ Ξ½β Ξ½β: Measure Ξ±}
{s t : Set Ξ±}
section IsFinit... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 41 | 44 | theorem not_isFiniteMeasure_iff : Β¬IsFiniteMeasure ΞΌ β ΞΌ Set.univ = β := by |
refine β¨fun h => ?_, fun h => fun h' => h'.measure_univ_lt_top.ne hβ©
by_contra h'
exact h β¨lt_top_iff_ne_top.mpr h'β©
| 1,388 |
import Mathlib.MeasureTheory.Measure.Restrict
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter Function MeasurableSpace ENNReal
variable {Ξ± Ξ² Ξ΄ ΞΉ : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace Ξ±} [MeasurableSpace Ξ²] {ΞΌ Ξ½ Ξ½β Ξ½β: Measure Ξ±}
{s t : Set Ξ±}
section IsFinit... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 65 | 72 | theorem measure_compl_le_add_of_le_add [IsFiniteMeasure ΞΌ] (hs : MeasurableSet s)
(ht : MeasurableSet t) {Ξ΅ : ββ₯0β} (h : ΞΌ s β€ ΞΌ t + Ξ΅) : ΞΌ tαΆ β€ ΞΌ sαΆ + Ξ΅ := by |
rw [measure_compl ht (measure_ne_top ΞΌ _), measure_compl hs (measure_ne_top ΞΌ _),
tsub_le_iff_right]
calc
ΞΌ univ = ΞΌ univ - ΞΌ s + ΞΌ s := (tsub_add_cancel_of_le <| measure_mono s.subset_univ).symm
_ β€ ΞΌ univ - ΞΌ s + (ΞΌ t + Ξ΅) := add_le_add_left h _
_ = _ := by rw [add_right_comm, add_assoc]
| 1,388 |
import Mathlib.MeasureTheory.Measure.Restrict
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter Function MeasurableSpace ENNReal
variable {Ξ± Ξ² Ξ΄ ΞΉ : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace Ξ±} [MeasurableSpace Ξ²] {ΞΌ Ξ½ Ξ½β Ξ½β: Measure Ξ±}
{s t : Set Ξ±}
section IsFinit... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 132 | 139 | theorem Measure.isFiniteMeasure_map {m : MeasurableSpace Ξ±} (ΞΌ : Measure Ξ±) [IsFiniteMeasure ΞΌ]
(f : Ξ± β Ξ²) : IsFiniteMeasure (ΞΌ.map f) := by |
by_cases hf : AEMeasurable f ΞΌ
Β· constructor
rw [map_apply_of_aemeasurable hf MeasurableSet.univ]
exact measure_lt_top ΞΌ _
Β· rw [map_of_not_aemeasurable hf]
exact MeasureTheory.isFiniteMeasureZero
| 1,388 |
import Mathlib.MeasureTheory.Measure.Restrict
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter Function MeasurableSpace ENNReal
variable {Ξ± Ξ² Ξ΄ ΞΉ : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace Ξ±} [MeasurableSpace Ξ²] {ΞΌ Ξ½ Ξ½β Ξ½β: Measure Ξ±}
{s t : Set Ξ±}
section NoAtoms... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 378 | 379 | theorem Measure.restrict_singleton' {a : Ξ±} : ΞΌ.restrict {a} = 0 := by |
simp only [measure_singleton, Measure.restrict_eq_zero]
| 1,388 |
import Mathlib.MeasureTheory.Measure.Restrict
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter Function MeasurableSpace ENNReal
variable {Ξ± Ξ² Ξ΄ ΞΉ : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace Ξ±} [MeasurableSpace Ξ²] {ΞΌ Ξ½ Ξ½β Ξ½β: Measure Ξ±}
{s t : Set Ξ±}
section NoAtoms... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 390 | 393 | theorem _root_.Set.Countable.measure_zero (h : s.Countable) (ΞΌ : Measure Ξ±) [NoAtoms ΞΌ] :
ΞΌ s = 0 := by |
rw [β biUnion_of_singleton s, measure_biUnion_null_iff h]
simp
| 1,388 |
import Mathlib.MeasureTheory.Measure.Restrict
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter Function MeasurableSpace ENNReal
variable {Ξ± Ξ² Ξ΄ ΞΉ : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace Ξ±} [MeasurableSpace Ξ²] {ΞΌ Ξ½ Ξ½β Ξ½β: Measure Ξ±}
{s t : Set Ξ±}
section NoAtoms... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 396 | 398 | theorem _root_.Set.Countable.ae_not_mem (h : s.Countable) (ΞΌ : Measure Ξ±) [NoAtoms ΞΌ] :
βα΅ x βΞΌ, x β s := by |
simpa only [ae_iff, Classical.not_not] using h.measure_zero ΞΌ
| 1,388 |
import Mathlib.MeasureTheory.Measure.Restrict
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter Function MeasurableSpace ENNReal
variable {Ξ± Ξ² Ξ΄ ΞΉ : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace Ξ±} [MeasurableSpace Ξ²] {ΞΌ Ξ½ Ξ½β Ξ½β: Measure Ξ±}
{s t : Set Ξ±}
| Mathlib/MeasureTheory/Measure/Typeclasses.lean | 491 | 498 | theorem ite_ae_eq_of_measure_zero {Ξ³} (f : Ξ± β Ξ³) (g : Ξ± β Ξ³) (s : Set Ξ±) [DecidablePred (Β· β s)]
(hs_zero : ΞΌ s = 0) :
(fun x => ite (x β s) (f x) (g x)) =α΅[ΞΌ] g := by |
have h_ss : sαΆ β { a : Ξ± | ite (a β s) (f a) (g a) = g a } := fun x hx => by
simp [(Set.mem_compl_iff _ _).mp hx]
refine measure_mono_null ?_ hs_zero
conv_rhs => rw [β compl_compl s]
rwa [Set.compl_subset_compl]
| 1,388 |
import Mathlib.MeasureTheory.Measure.Restrict
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter Function MeasurableSpace ENNReal
variable {Ξ± Ξ² Ξ΄ ΞΉ : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace Ξ±} [MeasurableSpace Ξ²] {ΞΌ Ξ½ Ξ½β Ξ½β: Measure Ξ±}
{s t : Set Ξ±}
theorem ite_ae_... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 501 | 508 | theorem ite_ae_eq_of_measure_compl_zero {Ξ³} (f : Ξ± β Ξ³) (g : Ξ± β Ξ³)
(s : Set Ξ±) [DecidablePred (Β· β s)] (hs_zero : ΞΌ sαΆ = 0) :
(fun x => ite (x β s) (f x) (g x)) =α΅[ΞΌ] f := by |
rw [β mem_ae_iff] at hs_zero
filter_upwards [hs_zero]
intros
split_ifs
rfl
| 1,388 |
import Mathlib.MeasureTheory.Measure.Typeclasses
#align_import measure_theory.decomposition.unsigned_hahn from "leanprover-community/mathlib"@"0f1becb755b3d008b242c622e248a70556ad19e6"
open Set Filter
open scoped Classical
open Topology ENNReal
namespace MeasureTheory
variable {Ξ± : Type*} [MeasurableSpace Ξ±] {... | Mathlib/MeasureTheory/Decomposition/UnsignedHahn.lean | 37 | 176 | theorem hahn_decomposition [IsFiniteMeasure ΞΌ] [IsFiniteMeasure Ξ½] :
β s,
MeasurableSet s β§
(β t, MeasurableSet t β t β s β Ξ½ t β€ ΞΌ t) β§ β t, MeasurableSet t β t β sαΆ β ΞΌ t β€ Ξ½ t := by |
let d : Set Ξ± β β := fun s => ((ΞΌ s).toNNReal : β) - (Ξ½ s).toNNReal
let c : Set β := d '' { s | MeasurableSet s }
let Ξ³ : β := sSup c
have hΞΌ : β s, ΞΌ s β β := measure_ne_top ΞΌ
have hΞ½ : β s, Ξ½ s β β := measure_ne_top Ξ½
have to_nnreal_ΞΌ : β s, ((ΞΌ s).toNNReal : ββ₯0β) = ΞΌ s := fun s => ENNReal.coe_toNNReal ... | 1,389 |
import Mathlib.MeasureTheory.Measure.Typeclasses
#align_import measure_theory.measure.sub from "leanprover-community/mathlib"@"562bbf524c595c153470e53d36c57b6f891cc480"
open Set
namespace MeasureTheory
namespace Measure
noncomputable instance instSub {Ξ± : Type*} [MeasurableSpace Ξ±] : Sub (Measure Ξ±) :=
β¨fun ... | Mathlib/MeasureTheory/Measure/Sub.lean | 71 | 97 | theorem sub_apply [IsFiniteMeasure Ξ½] (hβ : MeasurableSet s) (hβ : Ξ½ β€ ΞΌ) :
(ΞΌ - Ξ½) s = ΞΌ s - Ξ½ s := by |
-- We begin by defining `measure_sub`, which will be equal to `(ΞΌ - Ξ½)`.
let measure_sub : Measure Ξ± := MeasureTheory.Measure.ofMeasurable
(fun (t : Set Ξ±) (_ : MeasurableSet t) => ΞΌ t - Ξ½ t) (by simp)
(fun g h_meas h_disj β¦ by
simp only [measure_iUnion h_disj h_meas]
rw [ENNReal.tsum_sub _ (hβ... | 1,390 |
import Mathlib.MeasureTheory.Measure.Typeclasses
#align_import measure_theory.measure.sub from "leanprover-community/mathlib"@"562bbf524c595c153470e53d36c57b6f891cc480"
open Set
namespace MeasureTheory
namespace Measure
noncomputable instance instSub {Ξ± : Type*} [MeasurableSpace Ξ±] : Sub (Measure Ξ±) :=
β¨fun ... | Mathlib/MeasureTheory/Measure/Sub.lean | 100 | 102 | theorem sub_add_cancel_of_le [IsFiniteMeasure Ξ½] (hβ : Ξ½ β€ ΞΌ) : ΞΌ - Ξ½ + Ξ½ = ΞΌ := by |
ext1 s h_s_meas
rw [add_apply, sub_apply h_s_meas hβ, tsub_add_cancel_of_le (hβ s)]
| 1,390 |
import Mathlib.MeasureTheory.Measure.Typeclasses
#align_import measure_theory.measure.sub from "leanprover-community/mathlib"@"562bbf524c595c153470e53d36c57b6f891cc480"
open Set
namespace MeasureTheory
namespace Measure
noncomputable instance instSub {Ξ± : Type*} [MeasurableSpace Ξ±] : Sub (Measure Ξ±) :=
β¨fun ... | Mathlib/MeasureTheory/Measure/Sub.lean | 105 | 134 | theorem restrict_sub_eq_restrict_sub_restrict (h_meas_s : MeasurableSet s) :
(ΞΌ - Ξ½).restrict s = ΞΌ.restrict s - Ξ½.restrict s := by |
repeat rw [sub_def]
have h_nonempty : { d | ΞΌ β€ d + Ξ½ }.Nonempty := β¨ΞΌ, Measure.le_add_right le_rflβ©
rw [restrict_sInf_eq_sInf_restrict h_nonempty h_meas_s]
apply le_antisymm
Β· refine sInf_le_sInf_of_forall_exists_le ?_
intro Ξ½' h_Ξ½'_in
rw [mem_setOf_eq] at h_Ξ½'_in
refine β¨Ξ½'.restrict s, ?_, rest... | 1,390 |
import Mathlib.MeasureTheory.Measure.Typeclasses
#align_import measure_theory.measure.sub from "leanprover-community/mathlib"@"562bbf524c595c153470e53d36c57b6f891cc480"
open Set
namespace MeasureTheory
namespace Measure
noncomputable instance instSub {Ξ± : Type*} [MeasurableSpace Ξ±] : Sub (Measure Ξ±) :=
β¨fun ... | Mathlib/MeasureTheory/Measure/Sub.lean | 137 | 139 | theorem sub_apply_eq_zero_of_restrict_le_restrict (h_le : ΞΌ.restrict s β€ Ξ½.restrict s)
(h_meas_s : MeasurableSet s) : (ΞΌ - Ξ½) s = 0 := by |
rw [β restrict_apply_self, restrict_sub_eq_restrict_sub_restrict, sub_eq_zero_of_le] <;> simp [*]
| 1,390 |
import Mathlib.MeasureTheory.Constructions.Cylinders
import Mathlib.MeasureTheory.Measure.Typeclasses
open Set
namespace MeasureTheory
variable {ΞΉ : Type*} {Ξ± : ΞΉ β Type*} [β i, MeasurableSpace (Ξ± i)]
{P : β J : Finset ΞΉ, Measure (β j : J, Ξ± j)}
def IsProjectiveMeasureFamily (P : β J : Finset ΞΉ, Measure (β j ... | Mathlib/MeasureTheory/Constructions/Projective.lean | 143 | 150 | theorem unique [β i, IsFiniteMeasure (P i)]
(hΞΌ : IsProjectiveLimit ΞΌ P) (hΞ½ : IsProjectiveLimit Ξ½ P) :
ΞΌ = Ξ½ := by |
haveI : IsFiniteMeasure ΞΌ := hΞΌ.isFiniteMeasure
refine ext_of_generate_finite (measurableCylinders Ξ±) generateFrom_measurableCylinders.symm
isPiSystem_measurableCylinders (fun s hs β¦ ?_) (hΞΌ.measure_univ_unique hΞ½)
obtain β¨I, S, hS, rflβ© := (mem_measurableCylinders _).mp hs
rw [hΞΌ.measure_cylinder _ hS, hΞ½... | 1,391 |
import Mathlib.MeasureTheory.Measure.Typeclasses
open scoped ENNReal
namespace MeasureTheory
variable {Ξ± : Type*}
noncomputable
def Measure.trim {m m0 : MeasurableSpace Ξ±} (ΞΌ : @Measure Ξ± m0) (hm : m β€ m0) : @Measure Ξ± m :=
@OuterMeasure.toMeasure Ξ± m ΞΌ.toOuterMeasure (hm.trans (le_toOuterMeasure_caratheodory... | Mathlib/MeasureTheory/Measure/Trim.lean | 37 | 38 | theorem trim_eq_self [MeasurableSpace Ξ±] {ΞΌ : Measure Ξ±} : ΞΌ.trim le_rfl = ΞΌ := by |
simp [Measure.trim]
| 1,392 |
import Mathlib.MeasureTheory.Measure.Typeclasses
open scoped ENNReal
namespace MeasureTheory
variable {Ξ± : Type*}
noncomputable
def Measure.trim {m m0 : MeasurableSpace Ξ±} (ΞΌ : @Measure Ξ± m0) (hm : m β€ m0) : @Measure Ξ± m :=
@OuterMeasure.toMeasure Ξ± m ΞΌ.toOuterMeasure (hm.trans (le_toOuterMeasure_caratheodory... | Mathlib/MeasureTheory/Measure/Trim.lean | 43 | 45 | theorem toOuterMeasure_trim_eq_trim_toOuterMeasure (ΞΌ : Measure Ξ±) (hm : m β€ m0) :
@Measure.toOuterMeasure _ m (ΞΌ.trim hm) = @OuterMeasure.trim _ m ΞΌ.toOuterMeasure := by |
rw [Measure.trim, toMeasure_toOuterMeasure (ms := m)]
| 1,392 |
import Mathlib.MeasureTheory.Measure.Typeclasses
open scoped ENNReal
namespace MeasureTheory
variable {Ξ± : Type*}
noncomputable
def Measure.trim {m m0 : MeasurableSpace Ξ±} (ΞΌ : @Measure Ξ± m0) (hm : m β€ m0) : @Measure Ξ± m :=
@OuterMeasure.toMeasure Ξ± m ΞΌ.toOuterMeasure (hm.trans (le_toOuterMeasure_caratheodory... | Mathlib/MeasureTheory/Measure/Trim.lean | 49 | 50 | theorem zero_trim (hm : m β€ m0) : (0 : Measure Ξ±).trim hm = (0 : @Measure Ξ± m) := by |
simp [Measure.trim, @OuterMeasure.toMeasure_zero _ m]
| 1,392 |
import Mathlib.MeasureTheory.Measure.Typeclasses
open scoped ENNReal
namespace MeasureTheory
variable {Ξ± : Type*}
noncomputable
def Measure.trim {m m0 : MeasurableSpace Ξ±} (ΞΌ : @Measure Ξ± m0) (hm : m β€ m0) : @Measure Ξ± m :=
@OuterMeasure.toMeasure Ξ± m ΞΌ.toOuterMeasure (hm.trans (le_toOuterMeasure_caratheodory... | Mathlib/MeasureTheory/Measure/Trim.lean | 53 | 54 | theorem trim_measurableSet_eq (hm : m β€ m0) (hs : @MeasurableSet Ξ± m s) : ΞΌ.trim hm s = ΞΌ s := by |
rw [Measure.trim, toMeasure_apply (ms := m) _ _ hs, Measure.coe_toOuterMeasure]
| 1,392 |
import Mathlib.MeasureTheory.Measure.Typeclasses
open scoped ENNReal
namespace MeasureTheory
variable {Ξ± : Type*}
noncomputable
def Measure.trim {m m0 : MeasurableSpace Ξ±} (ΞΌ : @Measure Ξ± m0) (hm : m β€ m0) : @Measure Ξ± m :=
@OuterMeasure.toMeasure Ξ± m ΞΌ.toOuterMeasure (hm.trans (le_toOuterMeasure_caratheodory... | Mathlib/MeasureTheory/Measure/Trim.lean | 57 | 59 | theorem le_trim (hm : m β€ m0) : ΞΌ s β€ ΞΌ.trim hm s := by |
simp_rw [Measure.trim]
exact @le_toMeasure_apply _ m _ _ _
| 1,392 |
import Mathlib.MeasureTheory.Measure.Typeclasses
open scoped ENNReal
namespace MeasureTheory
variable {Ξ± : Type*}
noncomputable
def Measure.trim {m m0 : MeasurableSpace Ξ±} (ΞΌ : @Measure Ξ± m0) (hm : m β€ m0) : @Measure Ξ± m :=
@OuterMeasure.toMeasure Ξ± m ΞΌ.toOuterMeasure (hm.trans (le_toOuterMeasure_caratheodory... | Mathlib/MeasureTheory/Measure/Trim.lean | 86 | 90 | theorem trim_trim {mβ mβ : MeasurableSpace Ξ±} {hmββ : mβ β€ mβ} {hmβ : mβ β€ m0} :
(ΞΌ.trim hmβ).trim hmββ = ΞΌ.trim (hmββ.trans hmβ) := by |
refine @Measure.ext _ mβ _ _ (fun t ht => ?_)
rw [trim_measurableSet_eq hmββ ht, trim_measurableSet_eq (hmββ.trans hmβ) ht,
trim_measurableSet_eq hmβ (hmββ t ht)]
| 1,392 |
import Mathlib.MeasureTheory.Measure.Typeclasses
open scoped ENNReal
namespace MeasureTheory
variable {Ξ± : Type*}
noncomputable
def Measure.trim {m m0 : MeasurableSpace Ξ±} (ΞΌ : @Measure Ξ± m0) (hm : m β€ m0) : @Measure Ξ± m :=
@OuterMeasure.toMeasure Ξ± m ΞΌ.toOuterMeasure (hm.trans (le_toOuterMeasure_caratheodory... | Mathlib/MeasureTheory/Measure/Trim.lean | 93 | 98 | theorem restrict_trim (hm : m β€ m0) (ΞΌ : Measure Ξ±) (hs : @MeasurableSet Ξ± m s) :
@Measure.restrict Ξ± m (ΞΌ.trim hm) s = (ΞΌ.restrict s).trim hm := by |
refine @Measure.ext _ m _ _ (fun t ht => ?_)
rw [@Measure.restrict_apply Ξ± m _ _ _ ht, trim_measurableSet_eq hm ht,
Measure.restrict_apply (hm t ht),
trim_measurableSet_eq hm (@MeasurableSet.inter Ξ± m t s ht hs)]
| 1,392 |
import Mathlib.MeasureTheory.Measure.Typeclasses
open scoped ENNReal
namespace MeasureTheory
variable {Ξ± : Type*}
noncomputable
def Measure.trim {m m0 : MeasurableSpace Ξ±} (ΞΌ : @Measure Ξ± m0) (hm : m β€ m0) : @Measure Ξ± m :=
@OuterMeasure.toMeasure Ξ± m ΞΌ.toOuterMeasure (hm.trans (le_toOuterMeasure_caratheodory... | Mathlib/MeasureTheory/Measure/Trim.lean | 107 | 121 | theorem sigmaFiniteTrim_mono {m mβ m0 : MeasurableSpace Ξ±} {ΞΌ : Measure Ξ±} (hm : m β€ m0)
(hmβ : mβ β€ m) [SigmaFinite (ΞΌ.trim (hmβ.trans hm))] : SigmaFinite (ΞΌ.trim hm) := by |
refine β¨β¨?_β©β©
refine
{ set := spanningSets (ΞΌ.trim (hmβ.trans hm))
set_mem := fun _ => Set.mem_univ _
finite := fun i => ?_
spanning := iUnion_spanningSets _ }
calc
(ΞΌ.trim hm) (spanningSets (ΞΌ.trim (hmβ.trans hm)) i) =
((ΞΌ.trim hm).trim hmβ) (spanningSets (ΞΌ.trim (hmβ.trans hm)... | 1,392 |
import Mathlib.MeasureTheory.Measure.Typeclasses
open scoped ENNReal
namespace MeasureTheory
variable {Ξ± : Type*}
noncomputable
def Measure.trim {m m0 : MeasurableSpace Ξ±} (ΞΌ : @Measure Ξ± m0) (hm : m β€ m0) : @Measure Ξ± m :=
@OuterMeasure.toMeasure Ξ± m ΞΌ.toOuterMeasure (hm.trans (le_toOuterMeasure_caratheodory... | Mathlib/MeasureTheory/Measure/Trim.lean | 124 | 128 | theorem sigmaFinite_trim_bot_iff : SigmaFinite (ΞΌ.trim bot_le) β IsFiniteMeasure ΞΌ := by |
rw [sigmaFinite_bot_iff]
refine β¨fun h => β¨?_β©, fun h => β¨?_β©β© <;> have h_univ := h.measure_univ_lt_top
Β· rwa [trim_measurableSet_eq bot_le MeasurableSet.univ] at h_univ
Β· rwa [trim_measurableSet_eq bot_le MeasurableSet.univ]
| 1,392 |
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated
open Function Set
open scoped ENNReal Classical
noncomputable section
variable {Ξ± Ξ² Ξ΄ : Type*} [MeasurableSpace Ξ±] [MeasurableSpace Ξ²] {s : Set Ξ±} ... | Mathlib/MeasureTheory/Measure/Dirac.lean | 45 | 49 | theorem dirac_apply_of_mem {a : Ξ±} (h : a β s) : dirac a s = 1 := by |
have : β t : Set Ξ±, a β t β t.indicator (1 : Ξ± β ββ₯0β) a = 1 := fun t ht => indicator_of_mem ht 1
refine le_antisymm (this univ trivial βΈ ?_) (this s h βΈ le_dirac_apply)
rw [β dirac_apply' a MeasurableSet.univ]
exact measure_mono (subset_univ s)
| 1,393 |
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated
open Function Set
open scoped ENNReal Classical
noncomputable section
variable {Ξ± Ξ² Ξ΄ : Type*} [MeasurableSpace Ξ±] [MeasurableSpace Ξ²] {s : Set Ξ±} ... | Mathlib/MeasureTheory/Measure/Dirac.lean | 53 | 59 | theorem dirac_apply [MeasurableSingletonClass Ξ±] (a : Ξ±) (s : Set Ξ±) :
dirac a s = s.indicator 1 a := by |
by_cases h : a β s; Β· rw [dirac_apply_of_mem h, indicator_of_mem h, Pi.one_apply]
rw [indicator_of_not_mem h, β nonpos_iff_eq_zero]
calc
dirac a s β€ dirac a {a}αΆ := measure_mono (subset_compl_comm.1 <| singleton_subset_iff.2 h)
_ = 0 := by simp [dirac_apply' _ (measurableSet_singleton _).compl]
| 1,393 |
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated
open Function Set
open scoped ENNReal Classical
noncomputable section
variable {Ξ± Ξ² Ξ΄ : Type*} [MeasurableSpace Ξ±] [MeasurableSpace Ξ²] {s : Set Ξ±} ... | Mathlib/MeasureTheory/Measure/Dirac.lean | 77 | 83 | theorem restrict_singleton (ΞΌ : Measure Ξ±) (a : Ξ±) : ΞΌ.restrict {a} = ΞΌ {a} β’ dirac a := by |
ext1 s hs
by_cases ha : a β s
Β· have : s β© {a} = {a} := by simpa
simp [*]
Β· have : s β© {a} = β
:= inter_singleton_eq_empty.2 ha
simp [*]
| 1,393 |
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated
open Function Set
open scoped ENNReal Classical
noncomputable section
variable {Ξ± Ξ² Ξ΄ : Type*} [MeasurableSpace Ξ±] [MeasurableSpace Ξ²] {s : Set Ξ±} ... | Mathlib/MeasureTheory/Measure/Dirac.lean | 87 | 92 | theorem map_eq_sum [Countable Ξ²] [MeasurableSingletonClass Ξ²] (ΞΌ : Measure Ξ±) (f : Ξ± β Ξ²)
(hf : Measurable f) : ΞΌ.map f = sum fun b : Ξ² => ΞΌ (f β»ΒΉ' {b}) β’ dirac b := by |
ext s
have : β y β s, MeasurableSet (f β»ΒΉ' {y}) := fun y _ => hf (measurableSet_singleton _)
simp [β tsum_measure_preimage_singleton (to_countable s) this, *,
tsum_subtype s fun b => ΞΌ (f β»ΒΉ' {b}), β indicator_mul_right s fun b => ΞΌ (f β»ΒΉ' {b})]
| 1,393 |
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated
open Function Set
open scoped ENNReal Classical
noncomputable section
variable {Ξ± Ξ² Ξ΄ : Type*} [MeasurableSpace Ξ±] [MeasurableSpace Ξ²] {s : Set Ξ±} ... | Mathlib/MeasureTheory/Measure/Dirac.lean | 97 | 98 | theorem sum_smul_dirac [Countable Ξ±] [MeasurableSingletonClass Ξ±] (ΞΌ : Measure Ξ±) :
(sum fun a => ΞΌ {a} β’ dirac a) = ΞΌ := by | simpa using (map_eq_sum ΞΌ id measurable_id).symm
| 1,393 |
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated
open Function Set
open scoped ENNReal Classical
noncomputable section
variable {Ξ± Ξ² Ξ΄ : Type*} [MeasurableSpace Ξ±] [MeasurableSpace Ξ²] {s : Set Ξ±} ... | Mathlib/MeasureTheory/Measure/Dirac.lean | 103 | 110 | theorem tsum_indicator_apply_singleton [Countable Ξ±] [MeasurableSingletonClass Ξ±] (ΞΌ : Measure Ξ±)
(s : Set Ξ±) (hs : MeasurableSet s) : (β' x : Ξ±, s.indicator (fun x => ΞΌ {x}) x) = ΞΌ s :=
calc
(β' x : Ξ±, s.indicator (fun x => ΞΌ {x}) x) =
Measure.sum (fun a => ΞΌ {a} β’ Measure.dirac a) s := by |
simp only [Measure.sum_apply _ hs, Measure.smul_apply, smul_eq_mul, Measure.dirac_apply,
Set.indicator_apply, mul_ite, Pi.one_apply, mul_one, mul_zero]
_ = ΞΌ s := by rw [ΞΌ.sum_smul_dirac]
| 1,393 |
import Mathlib.MeasureTheory.Measure.Dirac
set_option autoImplicit true
open Set
open scoped ENNReal Classical
variable [MeasurableSpace Ξ±] [MeasurableSpace Ξ²] {s : Set Ξ±}
noncomputable section
namespace MeasureTheory.Measure
def count : Measure Ξ± :=
sum dirac
#align measure_theory.measure.count MeasureTheo... | Mathlib/MeasureTheory/Measure/Count.lean | 39 | 40 | theorem count_apply (hs : MeasurableSet s) : count s = β' i : s, 1 := by |
simp only [count, sum_apply, hs, dirac_apply', β tsum_subtype s (1 : Ξ± β ββ₯0β), Pi.one_apply]
| 1,394 |
import Mathlib.MeasureTheory.Measure.Dirac
set_option autoImplicit true
open Set
open scoped ENNReal Classical
variable [MeasurableSpace Ξ±] [MeasurableSpace Ξ²] {s : Set Ξ±}
noncomputable section
namespace MeasureTheory.Measure
def count : Measure Ξ± :=
sum dirac
#align measure_theory.measure.count MeasureTheo... | Mathlib/MeasureTheory/Measure/Count.lean | 44 | 44 | theorem count_empty : count (β
: Set Ξ±) = 0 := by | rw [count_apply MeasurableSet.empty, tsum_empty]
| 1,394 |
import Mathlib.MeasureTheory.Measure.Dirac
set_option autoImplicit true
open Set
open scoped ENNReal Classical
variable [MeasurableSpace Ξ±] [MeasurableSpace Ξ²] {s : Set Ξ±}
noncomputable section
namespace MeasureTheory.Measure
def count : Measure Ξ± :=
sum dirac
#align measure_theory.measure.count MeasureTheo... | Mathlib/MeasureTheory/Measure/Count.lean | 48 | 53 | theorem count_apply_finset' {s : Finset Ξ±} (s_mble : MeasurableSet (s : Set Ξ±)) :
count (βs : Set Ξ±) = s.card :=
calc
count (βs : Set Ξ±) = β' i : (βs : Set Ξ±), 1 := count_apply s_mble
_ = β i β s, 1 := s.tsum_subtype 1
_ = s.card := by | simp
| 1,394 |
import Mathlib.MeasureTheory.Measure.Dirac
set_option autoImplicit true
open Set
open scoped ENNReal Classical
variable [MeasurableSpace Ξ±] [MeasurableSpace Ξ²] {s : Set Ξ±}
noncomputable section
namespace MeasureTheory.Measure
def count : Measure Ξ± :=
sum dirac
#align measure_theory.measure.count MeasureTheo... | Mathlib/MeasureTheory/Measure/Count.lean | 62 | 65 | theorem count_apply_finite' {s : Set Ξ±} (s_fin : s.Finite) (s_mble : MeasurableSet s) :
count s = s_fin.toFinset.card := by |
simp [β
@count_apply_finset' _ _ s_fin.toFinset (by simpa only [Finite.coe_toFinset] using s_mble)]
| 1,394 |
import Mathlib.MeasureTheory.Measure.Dirac
set_option autoImplicit true
open Set
open scoped ENNReal Classical
variable [MeasurableSpace Ξ±] [MeasurableSpace Ξ²] {s : Set Ξ±}
noncomputable section
namespace MeasureTheory.Measure
def count : Measure Ξ± :=
sum dirac
#align measure_theory.measure.count MeasureTheo... | Mathlib/MeasureTheory/Measure/Count.lean | 68 | 69 | theorem count_apply_finite [MeasurableSingletonClass Ξ±] (s : Set Ξ±) (hs : s.Finite) :
count s = hs.toFinset.card := by | rw [β count_apply_finset, Finite.coe_toFinset]
| 1,394 |
import Mathlib.MeasureTheory.Measure.Dirac
set_option autoImplicit true
open Set
open scoped ENNReal Classical
variable [MeasurableSpace Ξ±] [MeasurableSpace Ξ²] {s : Set Ξ±}
noncomputable section
namespace MeasureTheory.Measure
def count : Measure Ξ± :=
sum dirac
#align measure_theory.measure.count MeasureTheo... | Mathlib/MeasureTheory/Measure/Count.lean | 73 | 80 | theorem count_apply_infinite (hs : s.Infinite) : count s = β := by |
refine top_unique (le_of_tendsto' ENNReal.tendsto_nat_nhds_top fun n => ?_)
rcases hs.exists_subset_card_eq n with β¨t, ht, rflβ©
calc
(t.card : ββ₯0β) = β i β t, 1 := by simp
_ = β' i : (t : Set Ξ±), 1 := (t.tsum_subtype 1).symm
_ β€ count (t : Set Ξ±) := le_count_apply
_ β€ count s := measure_mono ht
| 1,394 |
import Mathlib.MeasureTheory.Measure.Dirac
set_option autoImplicit true
open Set
open scoped ENNReal Classical
variable [MeasurableSpace Ξ±] [MeasurableSpace Ξ²] {s : Set Ξ±}
noncomputable section
namespace MeasureTheory.Measure
def count : Measure Ξ± :=
sum dirac
#align measure_theory.measure.count MeasureTheo... | Mathlib/MeasureTheory/Measure/Count.lean | 84 | 88 | theorem count_apply_eq_top' (s_mble : MeasurableSet s) : count s = β β s.Infinite := by |
by_cases hs : s.Finite
Β· simp [Set.Infinite, hs, count_apply_finite' hs s_mble]
Β· change s.Infinite at hs
simp [hs, count_apply_infinite]
| 1,394 |
import Mathlib.MeasureTheory.Measure.Dirac
set_option autoImplicit true
open Set
open scoped ENNReal Classical
variable [MeasurableSpace Ξ±] [MeasurableSpace Ξ²] {s : Set Ξ±}
noncomputable section
namespace MeasureTheory.Measure
def count : Measure Ξ± :=
sum dirac
#align measure_theory.measure.count MeasureTheo... | Mathlib/MeasureTheory/Measure/Count.lean | 92 | 96 | theorem count_apply_eq_top [MeasurableSingletonClass Ξ±] : count s = β β s.Infinite := by |
by_cases hs : s.Finite
Β· exact count_apply_eq_top' hs.measurableSet
Β· change s.Infinite at hs
simp [hs, count_apply_infinite]
| 1,394 |
import Mathlib.MeasureTheory.Measure.Dirac
set_option autoImplicit true
open Set
open scoped ENNReal Classical
variable [MeasurableSpace Ξ±] [MeasurableSpace Ξ²] {s : Set Ξ±}
noncomputable section
namespace MeasureTheory.Measure
def count : Measure Ξ± :=
sum dirac
#align measure_theory.measure.count MeasureTheo... | Mathlib/MeasureTheory/Measure/Count.lean | 115 | 119 | theorem empty_of_count_eq_zero' (s_mble : MeasurableSet s) (hsc : count s = 0) : s = β
:= by |
have hs : s.Finite := by
rw [β count_apply_lt_top' s_mble, hsc]
exact WithTop.zero_lt_top
simpa [count_apply_finite' hs s_mble] using hsc
| 1,394 |
import Mathlib.MeasureTheory.Measure.Dirac
set_option autoImplicit true
open Set
open scoped ENNReal Classical
variable [MeasurableSpace Ξ±] [MeasurableSpace Ξ²] {s : Set Ξ±}
noncomputable section
namespace MeasureTheory.Measure
def count : Measure Ξ± :=
sum dirac
#align measure_theory.measure.count MeasureTheo... | Mathlib/MeasureTheory/Measure/Count.lean | 122 | 126 | theorem empty_of_count_eq_zero [MeasurableSingletonClass Ξ±] (hsc : count s = 0) : s = β
:= by |
have hs : s.Finite := by
rw [β count_apply_lt_top, hsc]
exact WithTop.zero_lt_top
simpa [count_apply_finite _ hs] using hsc
| 1,394 |
import Mathlib.Probability.ConditionalProbability
import Mathlib.MeasureTheory.Measure.Count
#align_import probability.cond_count from "leanprover-community/mathlib"@"117e93f82b5f959f8193857370109935291f0cc4"
noncomputable section
open ProbabilityTheory
open MeasureTheory MeasurableSpace
namespace ProbabilityT... | Mathlib/Probability/CondCount.lean | 59 | 59 | theorem condCount_empty_meas : (condCount β
: Measure Ξ©) = 0 := by | simp [condCount]
| 1,395 |
import Mathlib.Probability.ConditionalProbability
import Mathlib.MeasureTheory.Measure.Count
#align_import probability.cond_count from "leanprover-community/mathlib"@"117e93f82b5f959f8193857370109935291f0cc4"
noncomputable section
open ProbabilityTheory
open MeasureTheory MeasurableSpace
namespace ProbabilityT... | Mathlib/Probability/CondCount.lean | 62 | 62 | theorem condCount_empty {s : Set Ξ©} : condCount s β
= 0 := by | simp
| 1,395 |
import Mathlib.Probability.ConditionalProbability
import Mathlib.MeasureTheory.Measure.Count
#align_import probability.cond_count from "leanprover-community/mathlib"@"117e93f82b5f959f8193857370109935291f0cc4"
noncomputable section
open ProbabilityTheory
open MeasureTheory MeasurableSpace
namespace ProbabilityT... | Mathlib/Probability/CondCount.lean | 65 | 67 | theorem finite_of_condCount_ne_zero {s t : Set Ξ©} (h : condCount s t β 0) : s.Finite := by |
by_contra hs'
simp [condCount, cond, Measure.count_apply_infinite hs'] at h
| 1,395 |
import Mathlib.Probability.ConditionalProbability
import Mathlib.MeasureTheory.Measure.Count
#align_import probability.cond_count from "leanprover-community/mathlib"@"117e93f82b5f959f8193857370109935291f0cc4"
noncomputable section
open ProbabilityTheory
open MeasureTheory MeasurableSpace
namespace ProbabilityT... | Mathlib/Probability/CondCount.lean | 70 | 76 | theorem condCount_univ [Fintype Ξ©] {s : Set Ξ©} :
condCount Set.univ s = Measure.count s / Fintype.card Ξ© := by |
rw [condCount, cond_apply _ MeasurableSet.univ, β ENNReal.div_eq_inv_mul, Set.univ_inter]
congr
rw [β Finset.coe_univ, Measure.count_apply, Finset.univ.tsum_subtype' fun _ => (1 : ENNReal)]
Β· simp [Finset.card_univ]
Β· exact (@Finset.coe_univ Ξ© _).symm βΈ MeasurableSet.univ
| 1,395 |
import Mathlib.Probability.ConditionalProbability
import Mathlib.MeasureTheory.Measure.Count
#align_import probability.cond_count from "leanprover-community/mathlib"@"117e93f82b5f959f8193857370109935291f0cc4"
noncomputable section
open ProbabilityTheory
open MeasureTheory MeasurableSpace
namespace ProbabilityT... | Mathlib/Probability/CondCount.lean | 81 | 86 | theorem condCount_isProbabilityMeasure {s : Set Ξ©} (hs : s.Finite) (hs' : s.Nonempty) :
IsProbabilityMeasure (condCount s) :=
{ measure_univ := by |
rw [condCount, cond_apply _ hs.measurableSet, Set.inter_univ, ENNReal.inv_mul_cancel]
Β· exact fun h => hs'.ne_empty <| Measure.empty_of_count_eq_zero h
Β· exact (Measure.count_apply_lt_top.2 hs).ne }
| 1,395 |
import Mathlib.Probability.ConditionalProbability
import Mathlib.MeasureTheory.Measure.Count
#align_import probability.cond_count from "leanprover-community/mathlib"@"117e93f82b5f959f8193857370109935291f0cc4"
noncomputable section
open ProbabilityTheory
open MeasureTheory MeasurableSpace
namespace ProbabilityT... | Mathlib/Probability/CondCount.lean | 89 | 95 | theorem condCount_singleton (Ο : Ξ©) (t : Set Ξ©) [Decidable (Ο β t)] :
condCount {Ο} t = if Ο β t then 1 else 0 := by |
rw [condCount, cond_apply _ (measurableSet_singleton Ο), Measure.count_singleton, inv_one,
one_mul]
split_ifs
Β· rw [(by simpa : ({Ο} : Set Ξ©) β© t = {Ο}), Measure.count_singleton]
Β· rw [(by simpa : ({Ο} : Set Ξ©) β© t = β
), Measure.count_empty]
| 1,395 |
import Mathlib.Probability.ConditionalProbability
import Mathlib.MeasureTheory.Measure.Count
#align_import probability.cond_count from "leanprover-community/mathlib"@"117e93f82b5f959f8193857370109935291f0cc4"
noncomputable section
open ProbabilityTheory
open MeasureTheory MeasurableSpace
namespace ProbabilityT... | Mathlib/Probability/CondCount.lean | 100 | 101 | theorem condCount_inter_self (hs : s.Finite) : condCount s (s β© t) = condCount s t := by |
rw [condCount, cond_inter_self _ hs.measurableSet]
| 1,395 |
import Mathlib.Probability.ConditionalProbability
import Mathlib.MeasureTheory.Measure.Count
#align_import probability.cond_count from "leanprover-community/mathlib"@"117e93f82b5f959f8193857370109935291f0cc4"
noncomputable section
open ProbabilityTheory
open MeasureTheory MeasurableSpace
namespace ProbabilityT... | Mathlib/Probability/CondCount.lean | 104 | 107 | theorem condCount_self (hs : s.Finite) (hs' : s.Nonempty) : condCount s s = 1 := by |
rw [condCount, cond_apply _ hs.measurableSet, Set.inter_self, ENNReal.inv_mul_cancel]
Β· exact fun h => hs'.ne_empty <| Measure.empty_of_count_eq_zero h
Β· exact (Measure.count_apply_lt_top.2 hs).ne
| 1,395 |
import Mathlib.Probability.ConditionalProbability
import Mathlib.MeasureTheory.Measure.Count
#align_import probability.cond_count from "leanprover-community/mathlib"@"117e93f82b5f959f8193857370109935291f0cc4"
noncomputable section
open ProbabilityTheory
open MeasureTheory MeasurableSpace
namespace ProbabilityT... | Mathlib/Probability/CondCount.lean | 110 | 115 | theorem condCount_eq_one_of (hs : s.Finite) (hs' : s.Nonempty) (ht : s β t) :
condCount s t = 1 := by |
haveI := condCount_isProbabilityMeasure hs hs'
refine eq_of_le_of_not_lt prob_le_one ?_
rw [not_lt, β condCount_self hs hs']
exact measure_mono ht
| 1,395 |
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