mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i ↦ Exists.choose ⋯
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
i : ι
x : α
hx : f i x = Exists.choose ⋯ x
hfx : x ∈ {x | C ≤ ‖f i x‖₊}
⊢ {x | C ≤ ‖g i x‖₊}.indicator (g i) x = {x | C ≤ ‖f i x‖₊}.indicator (f i) x

pp: case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i ↦ Exists.choose ⋯
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
i : ι
x : α
hx : f i x = Exists.choose ⋯ x
hfx : x ∉ {x | C ≤ ‖f i x‖₊}
⊢ {x | C ≤ ‖g i x‖₊}.indicator (g i) x = {x | C ≤ ‖f i x‖₊}.indicator (f i) x

Tactic:
·

Goal state after:
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i ↦ Exists.choose ⋯
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
i : ι
x : α
hx : f i x = Exists.choose ⋯ x
hfx : x ∈ {x | C ≤ ‖f i x‖₊}
⊢ {x | C ≤ ‖g i x‖₊}.indicator (g i) x = {x | C ≤ ‖f i x‖₊}.indicator (f i) x
Module: Mathlib.MeasureTheory.Measure.FiniteMeasure

Goal state before:
pp: Ω : Type u_1
inst✝ : MeasurableSpace Ω
μ ν : FiniteMeasure Ω
⊢ ⇑(μ + ν) = ⇑μ + ⇑ν

Tactic:
funext
  simp only [Pi.add_apply, ← ENNReal.coe_inj, ennreal_coeFn_eq_coeFn_toMeasure,
    ENNReal.coe_add]
  norm_cast

Goal state after:
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i ↦ Exists.choose ⋯
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
i : ι
x : α
hx : f i x = Exists.choose ⋯ x
hfx : x ∉ {x | C ≤ ‖f i x‖₊}
⊢ {x | C ≤ ‖g i x‖₊}.indicator (g i) x = {x | C ≤ ‖f i x‖₊}.indicator (f i) x

Tactic:
·

Goal state after:
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i ↦ Exists.choose ⋯
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
i : ι
x : α
hx : f i x = Exists.choose ⋯ x
hfx : x ∉ {x | C ≤ ‖f i x‖₊}
⊢ {x | C ≤ ‖g i x‖₊}.indicator (g i) x = {x | C ≤ ‖f i x‖₊}.indicator (f i) x
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Real

Goal state before:
pp: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ
hp : 1 ≤ p
f : α → E
s : Set α
hs : MeasurableSet s
hf : Integrable (fun x ↦ ‖f x‖ ^ p) μ
hp' : p ≠ 0
hm : ¬m ≤ m0
⊢ ∫ (x : α) in s, ‖μ[f | m] x‖ ^ p ∂μ ≤ ∫ (x : α) in s, ‖f x‖ ^ p ∂μ

pp: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ
hp : 1 ≤ p
f : α → E
s : Set α
hs : MeasurableSet s
hf : Integrable (fun x ↦ ‖f x‖ ^ p) μ
hp' : p ≠ 0
hm : m ≤ m0
⊢ ∫ (x : α) in s, ‖μ[f | m] x‖ ^ p ∂μ ≤ ∫ (x : α) in s, ‖f x‖ ^ p ∂μ

Tactic:
·

Goal state after:
case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ
hp : 1 ≤ p
f : α → E
s : Set α
hs : MeasurableSet s
hf : Integrable (fun x ↦ ‖f x‖ ^ p) μ
hp' : p ≠ 0
hm : ¬m ≤ m0
⊢ ∫ (x : α) in s, ‖μ[f | m] x‖ ^ p ∂μ ≤ ∫ (x : α) in s, ‖f x‖ ^ p ∂μ
Module: Mathlib.MeasureTheory.Measure.ProbabilityMeasure

Goal state before:
pp: case hf
α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
s✝ : Set (α × β)
u : Set α
Hu : MeasurableSet u
v : Set β
Hv : MeasurableSet v
Heq : u ×ˢ v = s✝
⊢ Measurable fun a ↦ ↑a.1 u

pp: case hg
α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
s✝ : Set (α × β)
u : Set α
Hu : MeasurableSet u
v : Set β
Hv : MeasurableSet v
Heq : u ×ˢ v = s✝
⊢ Measurable fun a ↦ ↑a.2 v

Tactic:
·

Goal state after:
case hf
α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
s✝ : Set (α × β)
u : Set α
Hu : MeasurableSet u
v : Set β
Hv : MeasurableSet v
Heq : u ×ˢ v = s✝
⊢ Measurable fun a ↦ ↑a.1 u
Module: Mathlib.MeasureTheory.Measure.ProbabilityMeasure

Goal state before:
pp: case hg
α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
s✝ : Set (α × β)
u : Set α
Hu : MeasurableSet u
v : Set β
Hv : MeasurableSet v
Heq : u ×ˢ v = s✝
⊢ Measurable fun a ↦ ↑a.2 v

Tactic:
·

Goal state after:
case hg
α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
s✝ : Set (α × β)
u : Set α
Hu : MeasurableSet u
v : Set β
Hv : MeasurableSet v
Heq : u ×ˢ v = s✝
⊢ Measurable fun a ↦ ↑a.2 v
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Real

Goal state before:
pp: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ
hp : 1 ≤ p
f : α → E
s : Set α
hs : MeasurableSet s
hf : Integrable (fun x ↦ ‖f x‖ ^ p) μ
hp' : p ≠ 0
hm : m ≤ m0
hsig : ¬SigmaFinite (μ.trim hm)
⊢ ∫ (x : α) in s, ‖μ[f | m] x‖ ^ p ∂μ ≤ ∫ (x : α) in s, ‖f x‖ ^ p ∂μ

pp: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ
hp : 1 ≤ p
f : α → E
s : Set α
hs : MeasurableSet s
hf : Integrable (fun x ↦ ‖f x‖ ^ p) μ
hp' : p ≠ 0
hm : m ≤ m0
hsig : SigmaFinite (μ.trim hm)
⊢ ∫ (x : α) in s, ‖μ[f | m] x‖ ^ p ∂μ ≤ ∫ (x : α) in s, ‖f x‖ ^ p ∂μ

Tactic:
·

Goal state after:
case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ
hp : 1 ≤ p
f : α → E
s : Set α
hs : MeasurableSet s
hf : Integrable (fun x ↦ ‖f x‖ ^ p) μ
hp' : p ≠ 0
hm : m ≤ m0
hsig : ¬SigmaFinite (μ.trim hm)
⊢ ∫ (x : α) in s, ‖μ[f | m] x‖ ^ p ∂μ ≤ ∫ (x : α) in s, ‖f x‖ ^ p ∂μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Real

Goal state before:
pp: case refine_1
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ
hp : 1 ≤ p
f : α → E
s : Set α
hs : MeasurableSet s
hf : Integrable (fun x ↦ ‖f x‖ ^ p) μ
hp' : p ≠ 0
hm : m ≤ m0
hsig : SigmaFinite (μ.trim hm)
⊢ 0 ≤ᵐ[μ.restrict s] fun x ↦ ‖μ[f | m] x‖ ^ p

pp: case refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ
hp : 1 ≤ p
f : α → E
s : Set α
hs : MeasurableSet s
hf : Integrable (fun x ↦ ‖f x‖ ^ p) μ
hp' : p ≠ 0
hm : m ≤ m0
hsig : SigmaFinite (μ.trim hm)
⊢ Integrable μ[fun x ↦ ‖f x‖ ^ p | m] (μ.restrict s)

pp: case refine_3
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ
hp : 1 ≤ p
f : α → E
s : Set α
hs : MeasurableSet s
hf : Integrable (fun x ↦ ‖f x‖ ^ p) μ
hp' : p ≠ 0
hm : m ≤ m0
hsig : SigmaFinite (μ.trim hm)
⊢ (fun x ↦ ‖μ[f | m] x‖ ^ p) ≤ᵐ[μ.restrict s] μ[fun x ↦ ‖f x‖ ^ p | m]

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ
hp : 1 ≤ p
f : α → E
s : Set α
hs : MeasurableSet s
hf : Integrable (fun x ↦ ‖f x‖ ^ p) μ
hp' : p ≠ 0
hm : m ≤ m0
hsig : SigmaFinite (μ.trim hm)
⊢ 0 ≤ᵐ[μ.restrict s] fun x ↦ ‖μ[f | m] x‖ ^ p
Module: Mathlib.MeasureTheory.Measure.FiniteMeasure

Goal state before:
pp: α : Type u_3
β : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
Heval : ∀ {u : Set α} {v : Set β}, MeasurableSet u → MeasurableSet v → Measurable fun a ↦ ↑a.1 u * ↑a.2 v
⊢ Measurable fun a ↦ ((↑a.1).prod ↑a.2) univ

pp: α : Type u_3
β : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
Heval : ∀ {u : Set α} {v : Set β}, MeasurableSet u → MeasurableSet v → Measurable fun a ↦ ↑a.1 u * ↑a.2 v
⊢ ∀ s ∈ image2 (fun x1 x2 ↦ x1 ×ˢ x2) {s | MeasurableSet s} {t | MeasurableSet t},
    Measurable fun a ↦ ((↑a.1).prod ↑a.2) s

Tactic:
·

Goal state after:
α : Type u_3
β : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
Heval : ∀ {u : Set α} {v : Set β}, MeasurableSet u → MeasurableSet v → Measurable fun a ↦ ↑a.1 u * ↑a.2 v
⊢ Measurable fun a ↦ ((↑a.1).prod ↑a.2) univ
Module: Mathlib.MeasureTheory.Measure.ProbabilityMeasure

Goal state before:
pp: case inl
Ω : Type u_1
inst✝ : Nonempty Ω
m0 : MeasurableSpace Ω
s : Set Ω
⊢ 0 s = mass 0 * (normalize 0) s

pp: case inr
Ω : Type u_1
inst✝ : Nonempty Ω
m0 : MeasurableSpace Ω
μ : FiniteMeasure Ω
s : Set Ω
h : μ ≠ 0
⊢ μ s = μ.mass * μ.normalize s

Tactic:
·

Goal state after:
case inl
Ω : Type u_1
inst✝ : Nonempty Ω
m0 : MeasurableSpace Ω
s : Set Ω
⊢ 0 s = mass 0 * (normalize 0) s
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Real

Goal state before:
pp: case refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ
hp : 1 ≤ p
f : α → E
s : Set α
hs : MeasurableSet s
hf : Integrable (fun x ↦ ‖f x‖ ^ p) μ
hp' : p ≠ 0
hm : m ≤ m0
hsig : SigmaFinite (μ.trim hm)
⊢ Integrable μ[fun x ↦ ‖f x‖ ^ p | m] (μ.restrict s)

pp: case refine_3
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ
hp : 1 ≤ p
f : α → E
s : Set α
hs : MeasurableSet s
hf : Integrable (fun x ↦ ‖f x‖ ^ p) μ
hp' : p ≠ 0
hm : m ≤ m0
hsig : SigmaFinite (μ.trim hm)
⊢ (fun x ↦ ‖μ[f | m] x‖ ^ p) ≤ᵐ[μ.restrict s] μ[fun x ↦ ‖f x‖ ^ p | m]

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ
hp : 1 ≤ p
f : α → E
s : Set α
hs : MeasurableSet s
hf : Integrable (fun x ↦ ‖f x‖ ^ p) μ
hp' : p ≠ 0
hm : m ≤ m0
hsig : SigmaFinite (μ.trim hm)
⊢ Integrable μ[fun x ↦ ‖f x‖ ^ p | m] (μ.restrict s)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Real

Goal state before:
pp: case refine_3
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ
hp : 1 ≤ p
f : α → E
s : Set α
hs : MeasurableSet s
hf : Integrable (fun x ↦ ‖f x‖ ^ p) μ
hp' : p ≠ 0
hm : m ≤ m0
hsig : SigmaFinite (μ.trim hm)
⊢ (fun x ↦ ‖μ[f | m] x‖ ^ p) ≤ᵐ[μ.restrict s] μ[fun x ↦ ‖f x‖ ^ p | m]

Tactic:
·

Goal state after:
case refine_3
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ
hp : 1 ≤ p
f : α → E
s : Set α
hs : MeasurableSet s
hf : Integrable (fun x ↦ ‖f x‖ ^ p) μ
hp' : p ≠ 0
hm : m ≤ m0
hsig : SigmaFinite (μ.trim hm)
⊢ (fun x ↦ ‖μ[f | m] x‖ ^ p) ≤ᵐ[μ.restrict s] μ[fun x ↦ ‖f x‖ ^ p | m]
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup β
p : ℝ≥0∞
κ : Type u_4
u : Filter κ
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
fn : ι → α → β
hUI : UnifIntegrable fn p μ
hfn : ∀ (i : ι), AEStronglyMeasurable (fn i) μ
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ : 0 < δ
hδ' :
  ∀ (i : ι) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (fn i)) p μ ≤ ENNReal.ofReal ε
x✝ : ↑{g | ∃ ni, ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ fn (ni n) x) u (𝓝 (g x))}
t : Set α
ht : MeasurableSet t
ht' : μ t ≤ ENNReal.ofReal δ
f : α → β
s : κ → ι
hs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ fn (s n) x) u (𝓝 (f x))
⊢ ∀ (n : κ), AEStronglyMeasurable (t.indicator (fn (s n))) μ

pp: case refine_2
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup β
p : ℝ≥0∞
κ : Type u_4
u : Filter κ
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
fn : ι → α → β
hUI : UnifIntegrable fn p μ
hfn : ∀ (i : ι), AEStronglyMeasurable (fn i) μ
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ : 0 < δ
hδ' :
  ∀ (i : ι) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (fn i)) p μ ≤ ENNReal.ofReal ε
x✝ : ↑{g | ∃ ni, ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ fn (ni n) x) u (𝓝 (g x))}
t : Set α
ht : MeasurableSet t
ht' : μ t ≤ ENNReal.ofReal δ
f : α → β
s : κ → ι
hs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ fn (s n) x) u (𝓝 (f x))
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun x_1 ↦ t.indicator (fn (s x_1)) x) u (𝓝 (t.indicator ((fun f ↦ ↑f) ⟨f, ⋯⟩) x))

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup β
p : ℝ≥0∞
κ : Type u_4
u : Filter κ
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
fn : ι → α → β
hUI : UnifIntegrable fn p μ
hfn : ∀ (i : ι), AEStronglyMeasurable (fn i) μ
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ : 0 < δ
hδ' :
  ∀ (i : ι) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (fn i)) p μ ≤ ENNReal.ofReal ε
x✝ : ↑{g | ∃ ni, ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ fn (ni n) x) u (𝓝 (g x))}
t : Set α
ht : MeasurableSet t
ht' : μ t ≤ ENNReal.ofReal δ
f : α → β
s : κ → ι
hs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ fn (s n) x) u (𝓝 (f x))
⊢ ∀ (n : κ), AEStronglyMeasurable (t.indicator (fn (s n))) μ
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup β
p : ℝ≥0∞
κ : Type u_4
u : Filter κ
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
fn : ι → α → β
hUI : UnifIntegrable fn p μ
hfn : ∀ (i : ι), AEStronglyMeasurable (fn i) μ
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ : 0 < δ
hδ' :
  ∀ (i : ι) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (fn i)) p μ ≤ ENNReal.ofReal ε
x✝ : ↑{g | ∃ ni, ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ fn (ni n) x) u (𝓝 (g x))}
t : Set α
ht : MeasurableSet t
ht' : μ t ≤ ENNReal.ofReal δ
f : α → β
s : κ → ι
hs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ fn (s n) x) u (𝓝 (f x))
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun x_1 ↦ t.indicator (fn (s x_1)) x) u (𝓝 (t.indicator ((fun f ↦ ↑f) ⟨f, ⋯⟩) x))

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup β
p : ℝ≥0∞
κ : Type u_4
u : Filter κ
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
fn : ι → α → β
hUI : UnifIntegrable fn p μ
hfn : ∀ (i : ι), AEStronglyMeasurable (fn i) μ
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ : 0 < δ
hδ' :
  ∀ (i : ι) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (fn i)) p μ ≤ ENNReal.ofReal ε
x✝ : ↑{g | ∃ ni, ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ fn (ni n) x) u (𝓝 (g x))}
t : Set α
ht : MeasurableSet t
ht' : μ t ≤ ENNReal.ofReal δ
f : α → β
s : κ → ι
hs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ fn (s n) x) u (𝓝 (f x))
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun x_1 ↦ t.indicator (fn (s x_1)) x) u (𝓝 (t.indicator ((fun f ↦ ↑f) ⟨f, ⋯⟩) x))
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup β
p : ℝ≥0∞
κ : Type u_4
u : Filter κ
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
fn : ι → α → β
hUI : UnifIntegrable fn p μ
hfn : ∀ (i : ι), AEStronglyMeasurable (fn i) μ
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ : 0 < δ
hδ' :
  ∀ (i : ι) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (fn i)) p μ ≤ ENNReal.ofReal ε
x✝ : ↑{g | ∃ ni, ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ fn (ni n) x) u (𝓝 (g x))}
t : Set α
ht : MeasurableSet t
ht' : μ t ≤ ENNReal.ofReal δ
f : α → β
s : κ → ι
hs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ fn (s n) x) u (𝓝 (f x))
a : α
ha : Tendsto (fun n ↦ fn (s n) a) u (𝓝 (f a))
memt : a ∈ t
⊢ Tendsto (fun x ↦ t.indicator (fn (s x)) a) u (𝓝 (t.indicator f a))

pp: case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup β
p : ℝ≥0∞
κ : Type u_4
u : Filter κ
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
fn : ι → α → β
hUI : UnifIntegrable fn p μ
hfn : ∀ (i : ι), AEStronglyMeasurable (fn i) μ
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ : 0 < δ
hδ' :
  ∀ (i : ι) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (fn i)) p μ ≤ ENNReal.ofReal ε
x✝ : ↑{g | ∃ ni, ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ fn (ni n) x) u (𝓝 (g x))}
t : Set α
ht : MeasurableSet t
ht' : μ t ≤ ENNReal.ofReal δ
f : α → β
s : κ → ι
hs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ fn (s n) x) u (𝓝 (f x))
a : α
ha : Tendsto (fun n ↦ fn (s n) a) u (𝓝 (f a))
memt : a ∉ t
⊢ Tendsto (fun x ↦ t.indicator (fn (s x)) a) u (𝓝 (t.indicator f a))

Tactic:
·

Goal state after:
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup β
p : ℝ≥0∞
κ : Type u_4
u : Filter κ
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
fn : ι → α → β
hUI : UnifIntegrable fn p μ
hfn : ∀ (i : ι), AEStronglyMeasurable (fn i) μ
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ : 0 < δ
hδ' :
  ∀ (i : ι) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (fn i)) p μ ≤ ENNReal.ofReal ε
x✝ : ↑{g | ∃ ni, ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ fn (ni n) x) u (𝓝 (g x))}
t : Set α
ht : MeasurableSet t
ht' : μ t ≤ ENNReal.ofReal δ
f : α → β
s : κ → ι
hs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ fn (s n) x) u (𝓝 (f x))
a : α
ha : Tendsto (fun n ↦ fn (s n) a) u (𝓝 (f a))
memt : a ∈ t
⊢ Tendsto (fun x ↦ t.indicator (fn (s x)) a) u (𝓝 (t.indicator f a))
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Real

Goal state before:
pp: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
s : Set α
hs : MeasurableSet s
f : α → E
hfint : ¬Integrable f μ
⊢ ∫ (x : α) in s, ‖μ[f | m] x‖ ∂μ ≤ ∫ (x : α) in s, ‖f x‖ ∂μ

pp: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
s : Set α
hs : MeasurableSet s
f : α → E
hfint : Integrable f μ
⊢ ∫ (x : α) in s, ‖μ[f | m] x‖ ∂μ ≤ ∫ (x : α) in s, ‖f x‖ ∂μ

Tactic:
·

Goal state after:
case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
s : Set α
hs : MeasurableSet s
f : α → E
hfint : ¬Integrable f μ
⊢ ∫ (x : α) in s, ‖μ[f | m] x‖ ∂μ ≤ ∫ (x : α) in s, ‖f x‖ ∂μ
Module: Mathlib.MeasureTheory.Measure.ProbabilityMeasure

Goal state before:
pp: case pos
Ω : Type u_1
inst✝² : Nonempty Ω
m0 : MeasurableSpace Ω
μ : FiniteMeasure Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_2
F : Filter γ
μs : γ → FiniteMeasure Ω
μs_lim : Tendsto (fun i ↦ (μs i).normalize) F (𝓝 μ.normalize)
mass_lim : Tendsto (fun i ↦ (μs i).mass) F (𝓝 μ.mass)
f : Ω →ᵇ ℝ≥0
h_mass : μ.mass = 0
⊢ Tendsto (fun i ↦ (μs i).testAgainstNN f) F (𝓝 (μ.testAgainstNN f))

pp: case neg
Ω : Type u_1
inst✝² : Nonempty Ω
m0 : MeasurableSpace Ω
μ : FiniteMeasure Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_2
F : Filter γ
μs : γ → FiniteMeasure Ω
μs_lim : Tendsto (fun i ↦ (μs i).normalize) F (𝓝 μ.normalize)
mass_lim : Tendsto (fun i ↦ (μs i).mass) F (𝓝 μ.mass)
f : Ω →ᵇ ℝ≥0
h_mass : ¬μ.mass = 0
⊢ Tendsto (fun i ↦ (μs i).testAgainstNN f) F (𝓝 (μ.testAgainstNN f))

Tactic:
·

Goal state after:
case pos
Ω : Type u_1
inst✝² : Nonempty Ω
m0 : MeasurableSpace Ω
μ : FiniteMeasure Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_2
F : Filter γ
μs : γ → FiniteMeasure Ω
μs_lim : Tendsto (fun i ↦ (μs i).normalize) F (𝓝 μ.normalize)
mass_lim : Tendsto (fun i ↦ (μs i).mass) F (𝓝 μ.mass)
f : Ω →ᵇ ℝ≥0
h_mass : μ.mass = 0
⊢ Tendsto (fun i ↦ (μs i).testAgainstNN f) F (𝓝 (μ.testAgainstNN f))
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup β
p : ℝ≥0∞
κ : Type u_4
u : Filter κ
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
fn : ι → α → β
hUI : UnifIntegrable fn p μ
hfn : ∀ (i : ι), AEStronglyMeasurable (fn i) μ
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ : 0 < δ
hδ' :
  ∀ (i : ι) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (fn i)) p μ ≤ ENNReal.ofReal ε
x✝ : ↑{g | ∃ ni, ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ fn (ni n) x) u (𝓝 (g x))}
t : Set α
ht : MeasurableSet t
ht' : μ t ≤ ENNReal.ofReal δ
f : α → β
s : κ → ι
hs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ fn (s n) x) u (𝓝 (f x))
a : α
ha : Tendsto (fun n ↦ fn (s n) a) u (𝓝 (f a))
memt : a ∉ t
⊢ Tendsto (fun x ↦ t.indicator (fn (s x)) a) u (𝓝 (t.indicator f a))

Tactic:
·

Goal state after:
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup β
p : ℝ≥0∞
κ : Type u_4
u : Filter κ
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
fn : ι → α → β
hUI : UnifIntegrable fn p μ
hfn : ∀ (i : ι), AEStronglyMeasurable (fn i) μ
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ : 0 < δ
hδ' :
  ∀ (i : ι) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (fn i)) p μ ≤ ENNReal.ofReal ε
x✝ : ↑{g | ∃ ni, ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ fn (ni n) x) u (𝓝 (g x))}
t : Set α
ht : MeasurableSet t
ht' : μ t ≤ ENNReal.ofReal δ
f : α → β
s : κ → ι
hs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ fn (s n) x) u (𝓝 (f x))
a : α
ha : Tendsto (fun n ↦ fn (s n) a) u (𝓝 (f a))
memt : a ∉ t
⊢ Tendsto (fun x ↦ t.indicator (fn (s x)) a) u (𝓝 (t.indicator f a))
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ∞
hf : ∀ (i : ι), MemLp (f i) p μ
val✝ : Fintype ι
hι : Nonempty ι
⊢ ∃ C, ∀ (i : ι), eLpNorm (f i) p μ ≤ ↑C

pp: case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ∞
hf : ∀ (i : ι), MemLp (f i) p μ
val✝ : Fintype ι
hι : ¬Nonempty ι
⊢ ∃ C, ∀ (i : ι), eLpNorm (f i) p μ ≤ ↑C

Tactic:
·

Goal state after:
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ∞
hf : ∀ (i : ι), MemLp (f i) p μ
val✝ : Fintype ι
hι : Nonempty ι
⊢ ∃ C, ∀ (i : ι), eLpNorm (f i) p μ ≤ ↑C
Module: Mathlib.MeasureTheory.Measure.FiniteMeasure

Goal state before:
pp: Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
f₁ f₂ : Ω →ᵇ ℝ≥0
⊢ |↑(μ.testAgainstNN f₁) - ↑(μ.testAgainstNN f₂)| ≤ ↑μ.mass * dist f₁ f₂

Tactic:
apply abs_le.mpr

Goal state after:
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
f₁ f₂ : Ω →ᵇ ℝ≥0
⊢ -(↑μ.mass * dist f₁ f₂) ≤ ↑(μ.testAgainstNN f₁) - ↑(μ.testAgainstNN f₂) ∧
    ↑(μ.testAgainstNN f₁) - ↑(μ.testAgainstNN f₂) ≤ ↑μ.mass * dist f₁ f₂