statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
is_Theta.is_bounded_under_le_iff (h : f' =Θ[l] g') :
is_bounded_under (≤) l (norm ∘ f') ↔ is_bounded_under (≤) l (norm ∘ g') | by simp only [← is_O_const_of_ne (one_ne_zero' ℝ), h.is_O_congr_left] | lemma | asymptotics.is_Theta.is_bounded_under_le_iff | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [
"one_ne_zero'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta.smul [normed_space 𝕜 E'] [normed_space 𝕜' F'] {f₁ : α → 𝕜} {f₂ : α → 𝕜'}
{g₁ : α → E'} {g₂ : α → F'} (hf : f₁ =Θ[l] f₂) (hg : g₁ =Θ[l] g₂) :
(λ x, f₁ x • g₁ x) =Θ[l] (λ x, f₂ x • g₂ x) | ⟨hf.1.smul hg.1, hf.2.smul hg.2⟩ | lemma | asymptotics.is_Theta.smul | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta.mul {f₁ f₂ : α → 𝕜} {g₁ g₂ : α → 𝕜'} (h₁ : f₁ =Θ[l] g₁) (h₂ : f₂ =Θ[l] g₂) :
(λ x, f₁ x * f₂ x) =Θ[l] (λ x, g₁ x * g₂ x) | h₁.smul h₂ | lemma | asymptotics.is_Theta.mul | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta.inv {f : α → 𝕜} {g : α → 𝕜'} (h : f =Θ[l] g) : (λ x, (f x)⁻¹) =Θ[l] (λ x, (g x)⁻¹) | ⟨h.2.inv_rev h.1.eq_zero_imp, h.1.inv_rev h.2.eq_zero_imp⟩ | lemma | asymptotics.is_Theta.inv | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta_inv {f : α → 𝕜} {g : α → 𝕜'} :
(λ x, (f x)⁻¹) =Θ[l] (λ x, (g x)⁻¹) ↔ f =Θ[l] g | ⟨λ h, by simpa only [inv_inv] using h.inv, is_Theta.inv⟩ | lemma | asymptotics.is_Theta_inv | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [
"inv_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta.div {f₁ f₂ : α → 𝕜} {g₁ g₂ : α → 𝕜'} (h₁ : f₁ =Θ[l] g₁) (h₂ : f₂ =Θ[l] g₂) :
(λ x, f₁ x / f₂ x) =Θ[l] (λ x, g₁ x / g₂ x) | by simpa only [div_eq_mul_inv] using h₁.mul h₂.inv | lemma | asymptotics.is_Theta.div | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta.pow {f : α → 𝕜} {g : α → 𝕜'} (h : f =Θ[l] g) (n : ℕ) :
(λ x, (f x) ^ n) =Θ[l] (λ x, (g x) ^ n) | ⟨h.1.pow n, h.2.pow n⟩ | lemma | asymptotics.is_Theta.pow | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta.zpow {f : α → 𝕜} {g : α → 𝕜'} (h : f =Θ[l] g) (n : ℤ) :
(λ x, (f x) ^ n) =Θ[l] (λ x, (g x) ^ n) | begin
cases n,
{ simpa only [zpow_of_nat] using h.pow _ },
{ simpa only [zpow_neg_succ_of_nat] using (h.pow _).inv }
end | lemma | asymptotics.is_Theta.zpow | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [
"zpow_neg_succ_of_nat",
"zpow_of_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta_const_const {c₁ : E''} {c₂ : F''} (h₁ : c₁ ≠ 0) (h₂ : c₂ ≠ 0) :
(λ x : α, c₁) =Θ[l] (λ x, c₂) | ⟨is_O_const_const _ h₂ _, is_O_const_const _ h₁ _⟩ | lemma | asymptotics.is_Theta_const_const | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta_const_const_iff [ne_bot l] {c₁ : E''} {c₂ : F''} :
(λ x : α, c₁) =Θ[l] (λ x, c₂) ↔ (c₁ = 0 ↔ c₂ = 0) | by simpa only [is_Theta, is_O_const_const_iff, ← iff_def] using iff.comm | lemma | asymptotics.is_Theta_const_const_iff | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [
"iff_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta_zero_left : (λ x, (0 : E')) =Θ[l] g'' ↔ g'' =ᶠ[l] 0 | by simp only [is_Theta, is_O_zero, is_O_zero_right_iff, true_and] | lemma | asymptotics.is_Theta_zero_left | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta_zero_right : f'' =Θ[l] (λ x, (0 : F')) ↔ f'' =ᶠ[l] 0 | is_Theta_comm.trans is_Theta_zero_left | lemma | asymptotics.is_Theta_zero_right | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta_const_smul_left [normed_space 𝕜 E'] {c : 𝕜} (hc : c ≠ 0) :
(λ x, c • f' x) =Θ[l] g ↔ f' =Θ[l] g | and_congr (is_O_const_smul_left hc) (is_O_const_smul_right hc) | lemma | asymptotics.is_Theta_const_smul_left | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta_const_smul_right [normed_space 𝕜 F'] {c : 𝕜} (hc : c ≠ 0) :
f =Θ[l] (λ x, c • g' x) ↔ f =Θ[l] g' | and_congr (is_O_const_smul_right hc) (is_O_const_smul_left hc) | lemma | asymptotics.is_Theta_const_smul_right | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta_const_mul_left {c : 𝕜} {f : α → 𝕜} (hc : c ≠ 0) :
(λ x, c * f x) =Θ[l] g ↔ f =Θ[l] g | by simpa only [← smul_eq_mul] using is_Theta_const_smul_left hc | lemma | asymptotics.is_Theta_const_mul_left | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta_const_mul_right {c : 𝕜} {g : α → 𝕜} (hc : c ≠ 0) :
f =Θ[l] (λ x, c * g x) ↔ f =Θ[l] g | by simpa only [← smul_eq_mul] using is_Theta_const_smul_right hc | lemma | asymptotics.is_Theta_const_mul_right | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_sum (f : ℝⁿ → E) (vol : ι →ᵇᵃ (E →L[ℝ] F)) (π : tagged_prepartition I) : F | ∑ J in π.boxes, vol J (f (π.tag J)) | def | box_integral.integral_sum | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | The integral sum of `f : ℝⁿ → E` over a tagged prepartition `π` w.r.t. box-additive volume `vol`
with codomain `E →L[ℝ] F` is the sum of `vol J (f (π.tag J))` over all boxes of `π`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integral_sum_bUnion_tagged (f : ℝⁿ → E) (vol : ι →ᵇᵃ (E →L[ℝ] F)) (π : prepartition I)
(πi : Π J, tagged_prepartition J) :
integral_sum f vol (π.bUnion_tagged πi) = ∑ J in π.boxes, integral_sum f vol (πi J) | begin
refine (π.sum_bUnion_boxes _ _).trans (sum_congr rfl $ λ J hJ, sum_congr rfl $ λ J' hJ', _),
rw π.tag_bUnion_tagged hJ hJ'
end | lemma | box_integral.integral_sum_bUnion_tagged | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_sum_bUnion_partition (f : ℝⁿ → E) (vol : ι →ᵇᵃ (E →L[ℝ] F))
(π : tagged_prepartition I) (πi : Π J, prepartition J) (hπi : ∀ J ∈ π, (πi J).is_partition) :
integral_sum f vol (π.bUnion_prepartition πi) = integral_sum f vol π | begin
refine (π.to_prepartition.sum_bUnion_boxes _ _).trans (sum_congr rfl $ λ J hJ, _),
calc ∑ J' in (πi J).boxes, vol J' (f (π.tag $ π.to_prepartition.bUnion_index πi J'))
= ∑ J' in (πi J).boxes, vol J' (f (π.tag J)) :
sum_congr rfl (λ J' hJ', by rw [prepartition.bUnion_index_of_mem _ hJ hJ'])
... = v... | lemma | box_integral.integral_sum_bUnion_partition | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [
"le_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_sum_inf_partition (f : ℝⁿ → E) (vol : ι →ᵇᵃ (E →L[ℝ] F))
(π : tagged_prepartition I) {π' : prepartition I} (h : π'.is_partition) :
integral_sum f vol (π.inf_prepartition π') = integral_sum f vol π | integral_sum_bUnion_partition f vol π _ $ λ J hJ, h.restrict (prepartition.le_of_mem _ hJ) | lemma | box_integral.integral_sum_inf_partition | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_sum_fiberwise {α} (g : box ι → α) (f : ℝⁿ → E)
(vol : ι →ᵇᵃ (E →L[ℝ] F)) (π : tagged_prepartition I) :
∑ y in π.boxes.image g, integral_sum f vol (π.filter (λ x, g x = y)) = integral_sum f vol π | π.to_prepartition.sum_fiberwise g (λ J, vol J (f $ π.tag J)) | lemma | box_integral.integral_sum_fiberwise | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_sum_sub_partitions (f : ℝⁿ → E) (vol : ι →ᵇᵃ (E →L[ℝ] F))
{π₁ π₂ : tagged_prepartition I} (h₁ : π₁.is_partition) (h₂ : π₂.is_partition) :
integral_sum f vol π₁ - integral_sum f vol π₂ =
∑ J in (π₁.to_prepartition ⊓ π₂.to_prepartition).boxes,
(vol J (f $ (π₁.inf_prepartition π₂.to_prepartition).ta... | begin
rw [← integral_sum_inf_partition f vol π₁ h₂,
← integral_sum_inf_partition f vol π₂ h₁, integral_sum, integral_sum,
finset.sum_sub_distrib],
simp only [inf_prepartition_to_prepartition, _root_.inf_comm]
end | lemma | box_integral.integral_sum_sub_partitions | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_sum_disj_union (f : ℝⁿ → E) (vol : ι →ᵇᵃ (E →L[ℝ] F))
{π₁ π₂ : tagged_prepartition I} (h : disjoint π₁.Union π₂.Union) :
integral_sum f vol (π₁.disj_union π₂ h) = integral_sum f vol π₁ + integral_sum f vol π₂ | begin
refine (prepartition.sum_disj_union_boxes h _).trans
(congr_arg2 (+) (sum_congr rfl $ λ J hJ, _) (sum_congr rfl $ λ J hJ, _)),
{ rw disj_union_tag_of_mem_left _ hJ },
{ rw disj_union_tag_of_mem_right _ hJ }
end | lemma | box_integral.integral_sum_disj_union | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [
"congr_arg2",
"disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_sum_add (f g : ℝⁿ → E) (vol : ι →ᵇᵃ (E →L[ℝ] F))
(π : tagged_prepartition I) :
integral_sum (f + g) vol π = integral_sum f vol π + integral_sum g vol π | by simp only [integral_sum, pi.add_apply, (vol _).map_add, finset.sum_add_distrib] | lemma | box_integral.integral_sum_add | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_sum_neg (f : ℝⁿ → E) (vol : ι →ᵇᵃ (E →L[ℝ] F))
(π : tagged_prepartition I) :
integral_sum (-f) vol π = -integral_sum f vol π | by simp only [integral_sum, pi.neg_apply, (vol _).map_neg, finset.sum_neg_distrib] | lemma | box_integral.integral_sum_neg | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_sum_smul (c : ℝ) (f : ℝⁿ → E) (vol : ι →ᵇᵃ (E →L[ℝ] F))
(π : tagged_prepartition I) :
integral_sum (c • f) vol π = c • integral_sum f vol π | by simp only [integral_sum, finset.smul_sum, pi.smul_apply, continuous_linear_map.map_smul] | lemma | box_integral.integral_sum_smul | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [
"continuous_linear_map.map_smul",
"finset.smul_sum",
"pi.smul_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_integral (I : box ι) (l : integration_params) (f : ℝⁿ → E) (vol : ι →ᵇᵃ (E →L[ℝ] F))
(y : F) : Prop | tendsto (integral_sum f vol) (l.to_filter_Union I ⊤) (𝓝 y) | def | box_integral.has_integral | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | The predicate `has_integral I l f vol y` says that `y` is the integral of `f` over `I` along `l`
w.r.t. volume `vol`. This means that integral sums of `f` tend to `𝓝 y` along
`box_integral.integration_params.to_filter_Union I ⊤`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integrable (I : box ι) (l : integration_params) (f : ℝⁿ → E) (vol : ι →ᵇᵃ (E →L[ℝ] F)) | ∃ y, has_integral I l f vol y | def | box_integral.integrable | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | A function is integrable if there exists a vector that satisfies the `has_integral`
predicate. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integral (I : box ι) (l : integration_params) (f : ℝⁿ → E) (vol : ι →ᵇᵃ (E →L[ℝ] F)) | if h : integrable I l f vol then h.some else 0 | def | box_integral.integral | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | The integral of a function `f` over a box `I` along a filter `l` w.r.t. a volume `vol`. Returns
zero on non-integrable functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_integral.tendsto (h : has_integral I l f vol y) :
tendsto (integral_sum f vol) (l.to_filter_Union I ⊤) (𝓝 y) | h | lemma | box_integral.has_integral.tendsto | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | Reinterpret `box_integral.has_integral` as `filter.tendsto`, e.g., dot-notation theorems
that are shadowed in the `box_integral.has_integral` namespace. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_integral_iff : has_integral I l f vol y ↔
∀ ε > (0 : ℝ), ∃ r : ℝ≥0 → ℝⁿ → Ioi (0 : ℝ), (∀ c, l.r_cond (r c)) ∧
∀ c π, l.mem_base_set I c (r c) π → is_partition π → dist (integral_sum f vol π) y ≤ ε | ((l.has_basis_to_filter_Union_top I).tendsto_iff nhds_basis_closed_ball).trans $
by simp [@forall_swap ℝ≥0 (tagged_prepartition I)] | lemma | box_integral.has_integral_iff | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [
"forall_swap"
] | The `ε`-`δ` definition of `box_integral.has_integral`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_integral_of_mul (a : ℝ) (h : ∀ ε : ℝ, 0 < ε →
∃ r: ℝ≥0 → ℝⁿ → Ioi (0 : ℝ), (∀ c, l.r_cond (r c)) ∧ ∀ c π, l.mem_base_set I c (r c) π →
is_partition π → dist (integral_sum f vol π) y ≤ a * ε) :
has_integral I l f vol y | begin
refine has_integral_iff.2 (λ ε hε, _),
rcases exists_pos_mul_lt hε a with ⟨ε', hε', ha⟩,
rcases h ε' hε' with ⟨r, hr, H⟩,
exact ⟨r, hr, λ c π hπ hπp, (H c π hπ hπp).trans ha.le⟩
end | lemma | box_integral.has_integral_of_mul | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [
"exists_pos_mul_lt"
] | Quite often it is more natural to prove an estimate of the form `a * ε`, not `ε` in the RHS of
`box_integral.has_integral_iff`, so we provide this auxiliary lemma. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integrable_iff_cauchy [complete_space F] :
integrable I l f vol ↔ cauchy ((l.to_filter_Union I ⊤).map (integral_sum f vol)) | cauchy_map_iff_exists_tendsto.symm | lemma | box_integral.integrable_iff_cauchy | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [
"cauchy",
"complete_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable_iff_cauchy_basis [complete_space F] :
integrable I l f vol ↔ ∀ ε > (0 : ℝ), ∃ r : ℝ≥0 → ℝⁿ → Ioi (0 : ℝ), (∀ c, l.r_cond (r c)) ∧
∀ c₁ c₂ π₁ π₂, l.mem_base_set I c₁ (r c₁) π₁ → π₁.is_partition → l.mem_base_set I c₂ (r c₂) π₂ →
π₂.is_partition → dist (integral_sum f vol π₁) (integral_sum f vol π₂)... | begin
rw [integrable_iff_cauchy, cauchy_map_iff',
(l.has_basis_to_filter_Union_top _).prod_self.tendsto_iff uniformity_basis_dist_le],
refine forall₂_congr (λ ε ε0, exists_congr $ λ r, _),
simp only [exists_prop, prod.forall, set.mem_Union, exists_imp_distrib,
prod_mk_mem_set_prod_eq, and_imp, mem_inter_i... | lemma | box_integral.integrable_iff_cauchy_basis | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [
"and_imp",
"cauchy_map_iff'",
"complete_space",
"exists_imp_distrib",
"exists_prop",
"forall₂_congr",
"set.mem_Union"
] | In a complete space, a function is integrable if and only if its integral sums form a Cauchy
net. Here we restate this fact in terms of `∀ ε > 0, ∃ r, ...`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_integral.mono {l₁ l₂ : integration_params} (h : has_integral I l₁ f vol y)
(hl : l₂ ≤ l₁) : has_integral I l₂ f vol y | h.mono_left $ integration_params.to_filter_Union_mono _ hl _ | lemma | box_integral.has_integral.mono | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable.has_integral (h : integrable I l f vol) :
has_integral I l f vol (integral I l f vol) | by { rw [integral, dif_pos h], exact classical.some_spec h } | lemma | box_integral.integrable.has_integral | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable.mono {l'} (h : integrable I l f vol) (hle : l' ≤ l) : integrable I l' f vol | ⟨_, h.has_integral.mono hle⟩ | lemma | box_integral.integrable.mono | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_integral.unique (h : has_integral I l f vol y) (h' : has_integral I l f vol y') :
y = y' | tendsto_nhds_unique h h' | lemma | box_integral.has_integral.unique | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [
"tendsto_nhds_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_integral.integrable (h : has_integral I l f vol y) : integrable I l f vol | ⟨_, h⟩ | lemma | box_integral.has_integral.integrable | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_integral.integral_eq (h : has_integral I l f vol y) :
integral I l f vol = y | h.integrable.has_integral.unique h | lemma | box_integral.has_integral.integral_eq | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_integral.add (h : has_integral I l f vol y) (h' : has_integral I l g vol y') :
has_integral I l (f + g) vol (y + y') | by simpa only [has_integral, ← integral_sum_add] using h.add h' | lemma | box_integral.has_integral.add | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable.add (hf : integrable I l f vol) (hg : integrable I l g vol) :
integrable I l (f + g) vol | (hf.has_integral.add hg.has_integral).integrable | lemma | box_integral.integrable.add | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_add (hf : integrable I l f vol) (hg : integrable I l g vol) :
integral I l (f + g) vol = integral I l f vol + integral I l g vol | (hf.has_integral.add hg.has_integral).integral_eq | lemma | box_integral.integral_add | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_integral.neg (hf : has_integral I l f vol y) : has_integral I l (-f) vol (-y) | by simpa only [has_integral, ← integral_sum_neg] using hf.neg | lemma | box_integral.has_integral.neg | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable.neg (hf : integrable I l f vol) : integrable I l (-f) vol | hf.has_integral.neg.integrable | lemma | box_integral.integrable.neg | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable.of_neg (hf : integrable I l (-f) vol) : integrable I l f vol | neg_neg f ▸ hf.neg | lemma | box_integral.integrable.of_neg | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable_neg : integrable I l (-f) vol ↔ integrable I l f vol | ⟨λ h, h.of_neg, λ h, h.neg⟩ | lemma | box_integral.integrable_neg | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_neg : integral I l (-f) vol = -integral I l f vol | if h : integrable I l f vol then h.has_integral.neg.integral_eq
else by rw [integral, integral, dif_neg h, dif_neg (mt integrable.of_neg h), neg_zero] | lemma | box_integral.integral_neg | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_integral.sub (h : has_integral I l f vol y) (h' : has_integral I l g vol y') :
has_integral I l (f - g) vol (y - y') | by simpa only [sub_eq_add_neg] using h.add h'.neg | lemma | box_integral.has_integral.sub | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable.sub (hf : integrable I l f vol) (hg : integrable I l g vol) :
integrable I l (f - g) vol | (hf.has_integral.sub hg.has_integral).integrable | lemma | box_integral.integrable.sub | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_sub (hf : integrable I l f vol) (hg : integrable I l g vol) :
integral I l (f - g) vol = integral I l f vol - integral I l g vol | (hf.has_integral.sub hg.has_integral).integral_eq | lemma | box_integral.integral_sub | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_integral_const (c : E) : has_integral I l (λ _, c) vol (vol I c) | tendsto_const_nhds.congr' $ (l.eventually_is_partition I).mono $ λ π hπ,
((vol.map ⟨λ g : E →L[ℝ] F, g c, rfl, λ _ _, rfl⟩).sum_partition_boxes le_top hπ).symm | lemma | box_integral.has_integral_const | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [
"le_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_const (c : E) : integral I l (λ _, c) vol = vol I c | (has_integral_const c).integral_eq | lemma | box_integral.integral_const | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable_const (c : E) : integrable I l (λ _, c) vol | ⟨_, has_integral_const c⟩ | lemma | box_integral.integrable_const | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_integral_zero : has_integral I l (λ _, (0:E)) vol 0 | by simpa only [← (vol I).map_zero] using has_integral_const (0 : E) | lemma | box_integral.has_integral_zero | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable_zero : integrable I l (λ _, (0:E)) vol | ⟨0, has_integral_zero⟩ | lemma | box_integral.integrable_zero | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_zero : integral I l (λ _, (0:E)) vol = 0 | has_integral_zero.integral_eq | lemma | box_integral.integral_zero | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_integral_sum {α : Type*} {s : finset α} {f : α → ℝⁿ → E} {g : α → F}
(h : ∀ i ∈ s, has_integral I l (f i) vol (g i)) :
has_integral I l (λ x, ∑ i in s, f i x) vol (∑ i in s, g i) | begin
induction s using finset.induction_on with a s ha ihs, { simp [has_integral_zero] },
simp only [finset.sum_insert ha], rw finset.forall_mem_insert at h,
exact h.1.add (ihs h.2)
end | lemma | box_integral.has_integral_sum | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [
"finset",
"finset.forall_mem_insert",
"finset.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_integral.smul (hf : has_integral I l f vol y) (c : ℝ) :
has_integral I l (c • f) vol (c • y) | by simpa only [has_integral, ← integral_sum_smul]
using (tendsto_const_nhds : tendsto _ _ (𝓝 c)).smul hf | lemma | box_integral.has_integral.smul | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable.smul (hf : integrable I l f vol) (c : ℝ) :
integrable I l (c • f) vol | (hf.has_integral.smul c).integrable | lemma | box_integral.integrable.smul | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable.of_smul {c : ℝ} (hf : integrable I l (c • f) vol) (hc : c ≠ 0) :
integrable I l f vol | by { convert hf.smul c⁻¹, ext x, simp only [pi.smul_apply, inv_smul_smul₀ hc] } | lemma | box_integral.integrable.of_smul | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [
"inv_smul_smul₀",
"pi.smul_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_smul (c : ℝ) : integral I l (λ x, c • f x) vol = c • integral I l f vol | begin
rcases eq_or_ne c 0 with rfl | hc, { simp only [zero_smul, integral_zero] },
by_cases hf : integrable I l f vol,
{ exact (hf.has_integral.smul c).integral_eq },
{ have : ¬integrable I l (λ x, c • f x) vol, from mt (λ h, h.of_smul hc) hf,
rw [integral, integral, dif_neg hf, dif_neg this, smul_zero] }
e... | lemma | box_integral.integral_smul | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [
"eq_or_ne",
"smul_zero",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_nonneg {g : ℝⁿ → ℝ} (hg : ∀ x ∈ I.Icc, 0 ≤ g x)
(μ : measure ℝⁿ) [is_locally_finite_measure μ] :
0 ≤ integral I l g μ.to_box_additive.to_smul | begin
by_cases hgi : integrable I l g μ.to_box_additive.to_smul,
{ refine ge_of_tendsto' hgi.has_integral (λ π, sum_nonneg $ λ J hJ, _),
exact mul_nonneg ennreal.to_real_nonneg (hg _ $ π.tag_mem_Icc _) },
{ rw [integral, dif_neg hgi] }
end | lemma | box_integral.integral_nonneg | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [
"ennreal.to_real_nonneg",
"ge_of_tendsto'"
] | The integral of a nonnegative function w.r.t. a volume generated by a locally-finite measure is
nonnegative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_integral_le_of_norm_le {g : ℝⁿ → ℝ} (hle : ∀ x ∈ I.Icc, ‖f x‖ ≤ g x)
(μ : measure ℝⁿ) [is_locally_finite_measure μ]
(hg : integrable I l g μ.to_box_additive.to_smul) :
‖(integral I l f μ.to_box_additive.to_smul : E)‖ ≤
integral I l g μ.to_box_additive.to_smul | begin
by_cases hfi : integrable.{u v v} I l f μ.to_box_additive.to_smul,
{ refine le_of_tendsto_of_tendsto' hfi.has_integral.norm hg.has_integral (λ π, _),
refine norm_sum_le_of_le _ (λ J hJ, _),
simp only [box_additive_map.to_smul_apply, norm_smul, smul_eq_mul, real.norm_eq_abs,
μ.to_box_additive_app... | lemma | box_integral.norm_integral_le_of_norm_le | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [
"abs_of_nonneg",
"ennreal.to_real_nonneg",
"le_of_tendsto_of_tendsto'",
"mul_le_mul_of_nonneg_left",
"norm_smul",
"real.norm_eq_abs",
"smul_eq_mul"
] | If `‖f x‖ ≤ g x` on `[l, u]` and `g` is integrable, then the norm of the integral of `f` is less
than or equal to the integral of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_integral_le_of_le_const {c : ℝ} (hc : ∀ x ∈ I.Icc, ‖f x‖ ≤ c)
(μ : measure ℝⁿ) [is_locally_finite_measure μ] :
‖(integral I l f μ.to_box_additive.to_smul : E)‖ ≤ (μ I).to_real * c | by simpa only [integral_const]
using norm_integral_le_of_norm_le hc μ (integrable_const c) | lemma | box_integral.norm_integral_le_of_le_const | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convergence_r (h : integrable I l f vol) (ε : ℝ) : ℝ≥0 → ℝⁿ → Ioi (0 : ℝ) | if hε : 0 < ε then (has_integral_iff.1 h.has_integral ε hε).some
else λ _ _, ⟨1, set.mem_Ioi.2 zero_lt_one⟩ | def | box_integral.integrable.convergence_r | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | If `ε > 0`, then `box_integral.integrable.convergence_r` is a function `r : ℝ≥0 → ℝⁿ → (0, ∞)`
such that for every `c : ℝ≥0`, for every tagged partition `π` subordinate to `r` (and satisfying
additional distortion estimates if `box_integral.integration_params.bDistortion l = tt`), the
corresponding integral sum is `ε`-... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convergence_r_cond (h : integrable I l f vol) (ε : ℝ) (c : ℝ≥0) :
l.r_cond (h.convergence_r ε c) | begin
rw convergence_r, split_ifs with h₀,
exacts [(has_integral_iff.1 h.has_integral ε h₀).some_spec.1 _, λ _ x, rfl]
end | lemma | box_integral.integrable.convergence_r_cond | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_integral_sum_integral_le_of_mem_base_set (h : integrable I l f vol) (h₀ : 0 < ε)
(hπ : l.mem_base_set I c (h.convergence_r ε c) π) (hπp : π.is_partition) :
dist (integral_sum f vol π) (integral I l f vol) ≤ ε | begin
rw [convergence_r, dif_pos h₀] at hπ,
exact (has_integral_iff.1 h.has_integral ε h₀).some_spec.2 c _ hπ hπp
end | lemma | box_integral.integrable.dist_integral_sum_integral_le_of_mem_base_set | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_integral_sum_le_of_mem_base_set (h : integrable I l f vol)
(hpos₁ : 0 < ε₁) (hpos₂ : 0 < ε₂) (h₁ : l.mem_base_set I c₁ (h.convergence_r ε₁ c₁) π₁)
(h₂ : l.mem_base_set I c₂ (h.convergence_r ε₂ c₂) π₂) (HU : π₁.Union = π₂.Union) :
dist (integral_sum f vol π₁) (integral_sum f vol π₂) ≤ ε₁ + ε₂ | begin
rcases h₁.exists_common_compl h₂ HU with ⟨π, hπU, hπc₁, hπc₂⟩,
set r : ℝⁿ → Ioi (0 : ℝ) := λ x, min (h.convergence_r ε₁ c₁ x) (h.convergence_r ε₂ c₂ x),
have hr : l.r_cond r := (h.convergence_r_cond _ c₁).min (h.convergence_r_cond _ c₂),
set πr := π.to_subordinate r,
have H₁ : dist (integral_sum f vol (... | lemma | box_integral.integrable.dist_integral_sum_le_of_mem_base_set | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [
"dist_triangle_right"
] | **Henstock-Sacks inequality**. Let `r₁ r₂ : ℝⁿ → (0, ∞)` be function such that for any tagged
*partition* of `I` subordinate to `rₖ`, `k=1,2`, the integral sum of `f` over this partition differs
from the integral of `f` by at most `εₖ`. Then for any two tagged *prepartition* `π₁ π₂` subordinate
to `r₁` and `r₂` respect... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_integral_sum_to_filter_prod_self_inf_Union_eq_uniformity (h : integrable I l f vol) :
tendsto
(λ π : tagged_prepartition I × tagged_prepartition I,
(integral_sum f vol π.1, integral_sum f vol π.2))
((l.to_filter I ×ᶠ l.to_filter I) ⊓ 𝓟 {π | π.1.Union = π.2.Union}) (𝓤 F) | begin
refine (((l.has_basis_to_filter I).prod_self.inf_principal _).tendsto_iff
uniformity_basis_dist_le).2 (λ ε ε0, _),
replace ε0 := half_pos ε0,
use [h.convergence_r (ε / 2), h.convergence_r_cond (ε / 2)], rintro ⟨π₁, π₂⟩ ⟨⟨h₁, h₂⟩, hU⟩,
rw ← add_halves ε,
exact h.dist_integral_sum_le_of_mem_base_set ε... | lemma | box_integral.integrable.tendsto_integral_sum_to_filter_prod_self_inf_Union_eq_uniformity | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [
"add_halves",
"half_pos"
] | If `f` is integrable on `I` along `l`, then for two sufficiently fine tagged prepartitions
(in the sense of the filter `box_integral.integration_params.to_filter l I`) such that they cover
the same part of `I`, the integral sums of `f` over `π₁` and `π₂` are very close to each other. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cauchy_map_integral_sum_to_filter_Union (h : integrable I l f vol) (π₀ : prepartition I) :
cauchy ((l.to_filter_Union I π₀).map (integral_sum f vol)) | begin
refine ⟨infer_instance, _⟩,
rw [prod_map_map_eq, ← to_filter_inf_Union_eq, ← prod_inf_prod, prod_principal_principal],
exact h.tendsto_integral_sum_to_filter_prod_self_inf_Union_eq_uniformity.mono_left
(inf_le_inf_left _ $ principal_mono.2 $ λ π h, h.1.trans h.2.symm)
end | lemma | box_integral.integrable.cauchy_map_integral_sum_to_filter_Union | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [
"cauchy",
"inf_le_inf_left"
] | If `f` is integrable on a box `I` along `l`, then for any fixed subset `s` of `I` that can be
represented as a finite union of boxes, the integral sums of `f` over tagged prepartitions that
cover exactly `s` form a Cauchy “sequence” along `l`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_subbox_aux (h : integrable I l f vol) (hJ : J ≤ I) :
∃ y : F, has_integral J l f vol y ∧
tendsto (integral_sum f vol) (l.to_filter_Union I (prepartition.single I J hJ)) (𝓝 y) | begin
refine (cauchy_map_iff_exists_tendsto.1
(h.cauchy_map_integral_sum_to_filter_Union (prepartition.single I J hJ))).imp (λ y hy, ⟨_, hy⟩),
convert hy.comp (l.tendsto_embed_box_to_filter_Union_top hJ) -- faster than `exact` here
end | lemma | box_integral.integrable.to_subbox_aux | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_subbox (h : integrable I l f vol) (hJ : J ≤ I) : integrable J l f vol | (h.to_subbox_aux hJ).imp $ λ y, and.left | lemma | box_integral.integrable.to_subbox | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | If `f` is integrable on a box `I`, then it is integrable on any subbox of `I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_integral_sum_to_filter_Union_single (h : integrable I l f vol) (hJ : J ≤ I) :
tendsto (integral_sum f vol) (l.to_filter_Union I (prepartition.single I J hJ))
(𝓝 $ integral J l f vol) | let ⟨y, h₁, h₂⟩ := h.to_subbox_aux hJ in h₁.integral_eq.symm ▸ h₂ | lemma | box_integral.integrable.tendsto_integral_sum_to_filter_Union_single | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | If `f` is integrable on a box `I`, then integral sums of `f` over tagged prepartitions
that cover exactly a subbox `J ≤ I` tend to the integral of `f` over `J` along `l`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_integral_sum_sum_integral_le_of_mem_base_set_of_Union_eq (h : integrable I l f vol)
(h0 : 0 < ε) (hπ : l.mem_base_set I c (h.convergence_r ε c) π) {π₀ : prepartition I}
(hU : π.Union = π₀.Union) :
dist (integral_sum f vol π) (∑ J in π₀.boxes, integral J l f vol) ≤ ε | begin
/- Let us prove that the distance is less than or equal to `ε + δ` for all positive `δ`. -/
refine le_of_forall_pos_le_add (λ δ δ0, _),
/- First we choose some constants. -/
set δ' : ℝ := δ / (π₀.boxes.card + 1),
have H0 : 0 < (π₀.boxes.card + 1 : ℝ) := nat.cast_add_one_pos _,
have δ'0 : 0 < δ' := div... | lemma | box_integral.integrable.dist_integral_sum_sum_integral_le_of_mem_base_set_of_Union_eq | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [
"dist_triangle",
"div_pos",
"finset.le_sup",
"le_rfl",
"nat.cast_add_one_pos",
"ring"
] | **Henstock-Sacks inequality**. Let `r : ℝⁿ → (0, ∞)` be a function such that for any tagged
*partition* of `I` subordinate to `r`, the integral sum of `f` over this partition differs from the
integral of `f` by at most `ε`. Then for any tagged *prepartition* `π` subordinate to `r`, the
integral sum of `f` over this pre... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_integral_sum_sum_integral_le_of_mem_base_set (h : integrable I l f vol)
(h0 : 0 < ε) (hπ : l.mem_base_set I c (h.convergence_r ε c) π) :
dist (integral_sum f vol π) (∑ J in π.boxes, integral J l f vol) ≤ ε | h.dist_integral_sum_sum_integral_le_of_mem_base_set_of_Union_eq h0 hπ rfl | lemma | box_integral.integrable.dist_integral_sum_sum_integral_le_of_mem_base_set | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | **Henstock-Sacks inequality**. Let `r : ℝⁿ → (0, ∞)` be a function such that for any tagged
*partition* of `I` subordinate to `r`, the integral sum of `f` over this partition differs from the
integral of `f` by at most `ε`. Then for any tagged *prepartition* `π` subordinate to `r`, the
integral sum of `f` over this pre... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_integral_sum_sum_integral (h : integrable I l f vol) (π₀ : prepartition I) :
tendsto (integral_sum f vol) (l.to_filter_Union I π₀) (𝓝 $ ∑ J in π₀.boxes, integral J l f vol) | begin
refine ((l.has_basis_to_filter_Union I π₀).tendsto_iff nhds_basis_closed_ball).2 (λ ε ε0, _),
refine ⟨h.convergence_r ε, h.convergence_r_cond ε, _⟩,
simp only [mem_inter_iff, set.mem_Union, mem_set_of_eq],
rintro π ⟨c, hc, hU⟩,
exact h.dist_integral_sum_sum_integral_le_of_mem_base_set_of_Union_eq ε0 hc ... | lemma | box_integral.integrable.tendsto_integral_sum_sum_integral | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [
"set.mem_Union"
] | Integral sum of `f` over a tagged prepartition `π` such that `π.Union = π₀.Union` tends to the
sum of integrals of `f` over the boxes of `π₀`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_integral_congr (h : integrable I l f vol) {π₁ π₂ : prepartition I}
(hU : π₁.Union = π₂.Union) :
∑ J in π₁.boxes, integral J l f vol = ∑ J in π₂.boxes, integral J l f vol | begin
refine tendsto_nhds_unique (h.tendsto_integral_sum_sum_integral π₁) _,
rw l.to_filter_Union_congr _ hU,
exact h.tendsto_integral_sum_sum_integral π₂
end | lemma | box_integral.integrable.sum_integral_congr | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [
"tendsto_nhds_unique"
] | If `f` is integrable on `I`, then `λ J, integral J l f vol` is box-additive on subboxes of `I`:
if `π₁`, `π₂` are two prepartitions of `I` covering the same part of `I`, then the sum of integrals
of `f` over the boxes of `π₁` is equal to the sum of integrals of `f` over the boxes of `π₂`.
See also `box_integral.integr... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_box_additive (h : integrable I l f vol) : ι →ᵇᵃ[I] F | { to_fun := λ J, integral J l f vol,
sum_partition_boxes' := λ J hJ π hπ,
begin
replace hπ := hπ.Union_eq, rw ← prepartition.Union_top at hπ,
rw [(h.to_subbox (with_top.coe_le_coe.1 hJ)).sum_integral_congr hπ,
prepartition.top_boxes, sum_singleton]
end } | def | box_integral.integrable.to_box_additive | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [] | If `f` is integrable on `I`, then `λ J, integral J l f vol` is box-additive on subboxes of `I`:
if `π₁`, `π₂` are two prepartitions of `I` covering the same part of `I`, then the sum of integrals
of `f` over the boxes of `π₁` is equal to the sum of integrals of `f` over the boxes of `π₂`.
See also `box_integral.integr... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integrable_of_continuous_on [complete_space E] {I : box ι} {f : ℝⁿ → E}
(hc : continuous_on f I.Icc) (μ : measure ℝⁿ) [is_locally_finite_measure μ] :
integrable.{u v v} I l f μ.to_box_additive.to_smul | begin
have huc := I.is_compact_Icc.uniform_continuous_on_of_continuous hc,
rw metric.uniform_continuous_on_iff_le at huc,
refine integrable_iff_cauchy_basis.2 (λ ε ε0, _),
rcases exists_pos_mul_lt ε0 (μ.to_box_additive I) with ⟨ε', ε0', hε⟩,
rcases huc ε' ε0' with ⟨δ, δ0 : 0 < δ, Hδ⟩,
refine ⟨λ _ _, ⟨δ / 2,... | lemma | box_integral.integrable_of_continuous_on | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [
"abs_of_nonneg",
"add_halves",
"complete_space",
"continuous_on",
"dist_triangle_left",
"ennreal.to_real_nonneg",
"exists_pos_mul_lt",
"finset.sum_mul",
"half_pos",
"le_top",
"metric.uniform_continuous_on_iff_le",
"mul_le_mul_of_nonneg_left",
"norm_smul",
"real.norm_eq_abs",
"smul_sub"
] | A continuous function is box-integrable with respect to any locally finite measure.
This is true for any volume with bounded variation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_integral_of_bRiemann_eq_ff_of_forall_is_o (hl : l.bRiemann = ff)
(B : ι →ᵇᵃ[I] ℝ) (hB0 : ∀ J, 0 ≤ B J) (g : ι →ᵇᵃ[I] F) (s : set ℝⁿ) (hs : s.countable)
(hlH : s.nonempty → l.bHenstock = tt)
(H₁ : ∀ (c : ℝ≥0) (x ∈ I.Icc ∩ s) (ε > (0 : ℝ)), ∃ δ > 0, ∀ J ≤ I,
J.Icc ⊆ metric.closed_ball x δ → x ∈ J.Icc →
... | begin
/- We choose `r x` differently for `x ∈ s` and `x ∉ s`.
For `x ∈ s`, we choose `εs` such that `∑' x : s, εs x < ε / 2 / 2 ^ #ι`, then choose `r x` so that
`dist (vol J (f x)) (g J) ≤ εs x` for `J` in the `r x`-neighborhood of `x`. This guarantees that
the sum of these distances over boxes `J` such that `... | lemma | box_integral.has_integral_of_bRiemann_eq_ff_of_forall_is_o | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [
"add_halves",
"div_pos",
"exists_pos_mul_lt",
"finset.card_le_of_subset",
"finset.coe_image",
"finset.le_sup",
"finset.mem_filter",
"fintype.card",
"half_pos",
"le_div_iff'",
"le_rfl",
"metric.closed_ball",
"metric.nhds_basis_closed_ball",
"mul_comm",
"nat.cast_pow",
"nat.cast_two",
... | This is an auxiliary lemma used to prove two statements at once. Use one of the next two
lemmas instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_integral_of_le_Henstock_of_forall_is_o (hl : l ≤ Henstock) (B : ι →ᵇᵃ[I] ℝ)
(hB0 : ∀ J, 0 ≤ B J) (g : ι →ᵇᵃ[I] F) (s : set ℝⁿ) (hs : s.countable)
(H₁ : ∀ (c : ℝ≥0) (x ∈ I.Icc ∩ s) (ε > (0 : ℝ)), ∃ δ > 0, ∀ J ≤ I,
J.Icc ⊆ metric.closed_ball x δ → x ∈ J.Icc → (l.bDistortion → J.distortion ≤ c) →
dist (vol... | have A : l.bHenstock, from hl.2.1.resolve_left dec_trivial,
has_integral_of_bRiemann_eq_ff_of_forall_is_o (hl.1.resolve_right dec_trivial) B hB0 _ s hs (λ _, A)
H₁ $ by simpa only [A, true_implies_iff] using H₂ | lemma | box_integral.has_integral_of_le_Henstock_of_forall_is_o | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [
"metric.closed_ball"
] | A function `f` has Henstock (or `⊥`) integral over `I` is equal to the value of a box-additive
function `g` on `I` provided that `vol J (f x)` is sufficiently close to `g J` for sufficiently
small boxes `J ∋ x`. This lemma is useful to prove, e.g., to prove the Divergence theorem for
integral along `⊥`.
Let `l` be eit... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_integral_McShane_of_forall_is_o (B : ι →ᵇᵃ[I] ℝ) (hB0 : ∀ J, 0 ≤ B J)
(g : ι →ᵇᵃ[I] F) (H : ∀ (c : ℝ≥0) (x ∈ I.Icc) (ε > (0 : ℝ)), ∃ δ > 0, ∀ J ≤ I,
J.Icc ⊆ metric.closed_ball x δ → dist (vol J (f x)) (g J) ≤ ε * B J) :
has_integral I McShane f vol (g I) | has_integral_of_bRiemann_eq_ff_of_forall_is_o rfl B hB0 g ∅ countable_empty (λ ⟨x, hx⟩, hx.elim)
(λ c x hx, hx.2.elim) $
by simpa only [McShane, coe_sort_ff, false_implies_iff, true_implies_iff, diff_empty] using H | lemma | box_integral.has_integral_McShane_of_forall_is_o | analysis.box_integral | src/analysis/box_integral/basic.lean | [
"analysis.box_integral.partition.filter",
"analysis.box_integral.partition.measure",
"topology.uniform_space.compact"
] | [
"metric.closed_ball"
] | Suppose that there exists a nonnegative box-additive function `B` with the following property.
For every `c : ℝ≥0`, a point `x ∈ I.Icc`, and a positive `ε` there exists `δ > 0` such that for any
box `J ≤ I` such that
- `J.Icc ⊆ metric.closed_ball x δ`;
- if `l.bDistortion` (i.e., `l = ⊥`), then the distortion of `J` ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_volume_sub_integral_face_upper_sub_lower_smul_le
{f : ℝⁿ⁺¹ → E} {f' : ℝⁿ⁺¹ →L[ℝ] E} (hfc : continuous_on f I.Icc)
{x : ℝⁿ⁺¹} (hxI : x ∈ I.Icc) {a : E} {ε : ℝ} (h0 : 0 < ε)
(hε : ∀ y ∈ I.Icc, ‖f y - a - f' (y - x)‖ ≤ ε * ‖y - x‖) {c : ℝ≥0} (hc : I.distortion ≤ c) :
‖(∏ j, (I.upper j - I.lower j)) • f' (pi.s... | begin
/- **Plan of the proof**. The difference of the integrals of the affine function
`λ y, a + f' (y - x)` over the faces `x i = I.upper i` and `x i = I.lower i` is equal to the
volume of `I` multiplied by `f' (pi.single i 1)`, so it suffices to show that the integral of
`f y - a - f' (y - x)` over each of th... | lemma | box_integral.norm_volume_sub_integral_face_upper_sub_lower_smul_le | analysis.box_integral | src/analysis/box_integral/divergence_theorem.lean | [
"analysis.box_integral.basic",
"analysis.box_integral.partition.additive",
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"continuous_on",
"fin.insert_nth_sub_same",
"mul_assoc",
"mul_le_mul_of_nonneg_left",
"mul_one",
"pi.single_smul'",
"smul_eq_mul",
"two_mul",
"zero_le_two"
] | Auxiliary lemma for the divergence theorem. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_integral_GP_pderiv (f : ℝⁿ⁺¹ → E) (f' : ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] E) (s : set ℝⁿ⁺¹)
(hs : s.countable) (Hs : ∀ x ∈ s, continuous_within_at f I.Icc x)
(Hd : ∀ x ∈ I.Icc \ s, has_fderiv_within_at f (f' x) I.Icc x) (i : fin (n + 1)) :
has_integral.{0 u u} I GP (λ x, f' x (pi.single i 1)) box_additive_map.volume
(int... | begin
/- Note that `f` is continuous on `I.Icc`, hence it is integrable on the faces of all boxes
`J ≤ I`, thus the difference of integrals over `x i = J.upper i` and `x i = J.lower i` is a
box-additive function of `J ≤ I`. -/
have Hc : continuous_on f I.Icc,
{ intros x hx,
by_cases hxs : x ∈ s,
exact... | lemma | box_integral.has_integral_GP_pderiv | analysis.box_integral | src/analysis/box_integral/divergence_theorem.lean | [
"analysis.box_integral.basic",
"analysis.box_integral.partition.additive",
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"Ioc_mem_nhds_within_Ioi",
"abs_nonneg",
"add_halves",
"continuous_on",
"continuous_within_at",
"dist_le_pi_dist",
"dist_triangle_right",
"ennreal.to_real_nonneg",
"exists_pos_mul_lt",
"ge_mem_nhds",
"half_pos",
"has_fderiv_within_at",
"le_abs_self",
"le_top",
"mul_le_mul_of_nonneg_right... | If `f : ℝⁿ⁺¹ → E` is differentiable on a closed rectangular box `I` with derivative `f'`, then
the partial derivative `λ x, f' x (pi.single i 1)` is Henstock-Kurzweil integrable with integral
equal to the difference of integrals of `f` over the faces `x i = I.upper i` and `x i = I.lower i`.
More precisely, we use a no... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_integral_GP_divergence_of_forall_has_deriv_within_at
(f : ℝⁿ⁺¹ → Eⁿ⁺¹) (f' : ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] Eⁿ⁺¹) (s : set ℝⁿ⁺¹) (hs : s.countable)
(Hs : ∀ x ∈ s, continuous_within_at f I.Icc x)
(Hd : ∀ x ∈ I.Icc \ s, has_fderiv_within_at f (f' x) I.Icc x) :
has_integral.{0 u u} I GP (λ x, ∑ i, f' x (pi.single i 1) i)
... | begin
refine has_integral_sum (λ i hi, _), clear hi,
simp only [has_fderiv_within_at_pi', continuous_within_at_pi] at Hd Hs,
convert has_integral_GP_pderiv I _ _ s hs (λ x hx, Hs x hx i) (λ x hx, Hd x hx i) i
end | lemma | box_integral.has_integral_GP_divergence_of_forall_has_deriv_within_at | analysis.box_integral | src/analysis/box_integral/divergence_theorem.lean | [
"analysis.box_integral.basic",
"analysis.box_integral.partition.additive",
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"continuous_within_at",
"continuous_within_at_pi",
"has_fderiv_within_at",
"has_fderiv_within_at_pi'"
] | Divergence theorem for a Henstock-Kurzweil style integral.
If `f : ℝⁿ⁺¹ → Eⁿ⁺¹` is differentiable on a closed rectangular box `I` with derivative `f'`, then
the divergence `∑ i, f' x (pi.single i 1) i` is Henstock-Kurzweil integrable with integral equal to
the sum of integrals of `f` over the faces of `I` taken with a... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_integral_indicator_const (l : integration_params) (hl : l.bRiemann = ff)
{s : set (ι → ℝ)} (hs : measurable_set s) (I : box ι) (y : E)
(μ : measure (ι → ℝ)) [is_locally_finite_measure μ] :
has_integral.{u v v} I l (s.indicator (λ _, y)) μ.to_box_additive.to_smul
((μ (s ∩ I)).to_real • y) | begin
refine has_integral_of_mul (‖y‖) (λ ε ε0, _),
lift ε to ℝ≥0 using ε0.le, rw nnreal.coe_pos at ε0,
/- First we choose a closed set `F ⊆ s ∩ I.Icc` and an open set `U ⊇ s` such that
both `(s ∩ I.Icc) \ F` and `U \ s` have measuer less than `ε`. -/
have A : μ (s ∩ I.Icc) ≠ ∞,
from ((measure_mono $ set.... | lemma | box_integral.has_integral_indicator_const | analysis.box_integral | src/analysis/box_integral/integrability.lean | [
"analysis.box_integral.basic",
"measure_theory.integral.set_integral",
"measure_theory.measure.regular"
] | [
"and_imp",
"ennreal.le_to_real_sub",
"ennreal.to_real_le_coe_of_le_coe",
"exists_prop",
"is_closed",
"is_open",
"le_rfl",
"lift",
"measurable_set",
"mul_comm",
"mul_le_mul_of_nonneg_right",
"nnreal.coe_pos",
"norm_smul",
"real.norm_eq_abs",
"set.inter_subset_right",
"set.mem_Union",
... | The indicator function of a measurable set is McShane integrable with respect to any
locally-finite measure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_integral_zero_of_ae_eq_zero {l : integration_params} {I : box ι} {f : (ι → ℝ) → E}
{μ : measure (ι → ℝ)} [is_locally_finite_measure μ] (hf : f =ᵐ[μ.restrict I] 0)
(hl : l.bRiemann = ff) :
has_integral.{u v v} I l f μ.to_box_additive.to_smul 0 | begin
/- Each set `{x | n < ‖f x‖ ≤ n + 1}`, `n : ℕ`, has measure zero. We cover it by an open set of
measure less than `ε / 2 ^ n / (n + 1)`. Then the norm of the integral sum is less than `ε`. -/
refine has_integral_iff.2 (λ ε ε0, _),
lift ε to ℝ≥0 using ε0.lt.le, rw [gt_iff_lt, nnreal.coe_pos] at ε0,
rcase... | lemma | box_integral.has_integral_zero_of_ae_eq_zero | analysis.box_integral | src/analysis/box_integral/integrability.lean | [
"analysis.box_integral.basic",
"measure_theory.integral.set_integral",
"measure_theory.measure.regular"
] | [
"abs_of_nonneg",
"ennreal.coe_le_coe",
"ennreal.coe_mul",
"ennreal.coe_nat",
"ennreal.coe_to_real",
"ennreal.div_zero",
"ennreal.mul_div_le",
"ennreal.to_real_nonneg",
"exists_prop",
"gt_iff_lt",
"is_open",
"lift",
"measurable_set",
"mul_comm",
"mul_le_mul_left'",
"mul_le_mul_of_nonneg... | If `f` is a.e. equal to zero on a rectangular box, then it has McShane integral zero on this
box. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_integral.congr_ae {l : integration_params} {I : box ι} {y : E} {f g : (ι → ℝ) → E}
{μ : measure (ι → ℝ)} [is_locally_finite_measure μ]
(hf : has_integral.{u v v} I l f μ.to_box_additive.to_smul y)
(hfg : f =ᵐ[μ.restrict I] g) (hl : l.bRiemann = ff) :
has_integral.{u v v} I l g μ.to_box_additive.to_smul y | begin
have : (g - f) =ᵐ[μ.restrict I] 0, from hfg.mono (λ x hx, sub_eq_zero.2 hx.symm),
simpa using hf.add (has_integral_zero_of_ae_eq_zero this hl)
end | lemma | box_integral.has_integral.congr_ae | analysis.box_integral | src/analysis/box_integral/integrability.lean | [
"analysis.box_integral.basic",
"measure_theory.integral.set_integral",
"measure_theory.measure.regular"
] | [] | If `f` has integral `y` on a box `I` with respect to a locally finite measure `μ` and `g` is
a.e. equal to `f` on `I`, then `g` has the same integral on `I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_box_integral (f : simple_func (ι → ℝ) E) (μ : measure (ι → ℝ))
[is_locally_finite_measure μ] (I : box ι) (l : integration_params) (hl : l.bRiemann = ff) :
has_integral.{u v v} I l f μ.to_box_additive.to_smul (f.integral (μ.restrict I)) | begin
induction f using measure_theory.simple_func.induction with y s hs f g hd hfi hgi,
{ simpa only [measure.restrict_apply hs, const_zero, integral_piecewise_zero, integral_const,
measure.restrict_apply, measurable_set.univ, set.univ_inter]
using box_integral.has_integral_indicator_const l hl hs I y ... | lemma | measure_theory.simple_func.has_box_integral | analysis.box_integral | src/analysis/box_integral/integrability.lean | [
"analysis.box_integral.basic",
"measure_theory.integral.set_integral",
"measure_theory.measure.regular"
] | [
"box_integral.has_integral_indicator_const",
"measurable_set.univ",
"measure_theory.simple_func.induction",
"set.univ_inter"
] | A simple function is McShane integrable w.r.t. any locally finite measure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
box_integral_eq_integral (f : simple_func (ι → ℝ) E) (μ : measure (ι → ℝ))
[is_locally_finite_measure μ] (I : box ι) (l : integration_params) (hl : l.bRiemann = ff) :
box_integral.integral.{u v v} I l f μ.to_box_additive.to_smul = f.integral (μ.restrict I) | (f.has_box_integral μ I l hl).integral_eq | lemma | measure_theory.simple_func.box_integral_eq_integral | analysis.box_integral | src/analysis/box_integral/integrability.lean | [
"analysis.box_integral.basic",
"measure_theory.integral.set_integral",
"measure_theory.measure.regular"
] | [] | For a simple function, its McShane (or Henstock, or `⊥`) box integral is equal to its
integral in the sense of `measure_theory.simple_func.integral`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integrable_on.has_box_integral [complete_space E] {f : (ι → ℝ) → E} {μ : measure (ι → ℝ)}
[is_locally_finite_measure μ] {I : box ι} (hf : integrable_on f I μ) (l : integration_params)
(hl : l.bRiemann = ff) :
has_integral.{u v v} I l f μ.to_box_additive.to_smul (∫ x in I, f x ∂ μ) | begin
borelize E,
/- First we replace an `ae_strongly_measurable` function by a measurable one. -/
rcases hf.ae_strongly_measurable with ⟨g, hg, hfg⟩,
haveI : separable_space (range g ∪ {0} : set E) := hg.separable_space_range_union_singleton,
rw integral_congr_ae hfg, have hgi : integrable_on g I μ := (integ... | lemma | measure_theory.integrable_on.has_box_integral | analysis.box_integral | src/analysis/box_integral/integrability.lean | [
"analysis.box_integral.basic",
"measure_theory.integral.set_integral",
"measure_theory.measure.regular"
] | [
"abs_of_nonneg",
"box_integral.has_integral.congr_ae",
"complete_space",
"dist_nndist",
"dist_triangle4",
"edist_nndist",
"ennreal.coe_le_coe",
"ennreal.to_real_nonneg",
"ge_mem_nhds",
"le_rfl",
"lift",
"mul_le_mul_of_nonneg_left",
"nnreal.coe_le_coe",
"nnreal.coe_pos",
"nnreal.exists_po... | If `f : ℝⁿ → E` is Bochner integrable w.r.t. a locally finite measure `μ` on a rectangular box
`I`, then it is McShane integrable on `I` with the same integral. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
box (ι : Type*) | (lower upper : ι → ℝ)
(lower_lt_upper : ∀ i, lower i < upper i) | structure | box_integral.box | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [] | A nontrivial rectangular box in `ι → ℝ` with corners `lower` and `upper`. Repesents the product
of half-open intervals `(lower i, upper i]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lower_le_upper : I.lower ≤ I.upper | λ i, (I.lower_lt_upper i).le | lemma | box_integral.box.lower_le_upper | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_ne_upper (i) : I.lower i ≠ I.upper i | (I.lower_lt_upper i).ne | lemma | box_integral.box.lower_ne_upper | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_mk {l u x : ι → ℝ} {H} : x ∈ mk l u H ↔ ∀ i, x i ∈ Ioc (l i) (u i) | iff.rfl | lemma | box_integral.box.mem_mk | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_coe : x ∈ (I : set (ι → ℝ)) ↔ x ∈ I | iff.rfl | lemma | box_integral.box.mem_coe | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_def : x ∈ I ↔ ∀ i, x i ∈ Ioc (I.lower i) (I.upper i) | iff.rfl | lemma | box_integral.box.mem_def | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_univ_Ioc {I : box ι} : x ∈ pi univ (λ i, Ioc (I.lower i) (I.upper i)) ↔ x ∈ I | mem_univ_pi | lemma | box_integral.box.mem_univ_Ioc | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_eq_pi : (I : set (ι → ℝ)) = pi univ (λ i, Ioc (I.lower i) (I.upper i)) | set.ext $ λ x, mem_univ_Ioc.symm | lemma | box_integral.box.coe_eq_pi | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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