-- Mathlib/Analysis/Convex/Integral.lean
theorem Convex.integral_mem [IsProbabilityMeasure μ] (hs : Convex ℝ s) (hsc : IsClosed s)
    (hf : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) : (∫ x, f x ∂μ) ∈ s

Proof state:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
s : Set E
f : α → E
inst✝ : IsProbabilityMeasure μ
hs : Convex ℝ s
hsc : IsClosed s
hf : ∀ᵐ (x : α) ∂μ, f x ∈ s
hfi : Integrable f μ
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
⊢ ∫ (x : α), f x ∂μ ∈ s

Tactic:
rcases hfi.aestronglyMeasurable with ⟨g, hgm, hfg⟩

Result:
case intro.intro
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
s : Set E
f : α → E
inst✝ : IsProbabilityMeasure μ
hs : Convex ℝ s
hsc : IsClosed s
hf : ∀ᵐ (x : α) ∂μ, f x ∈ s
hfi : Integrable f μ
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : α → E
hgm : StronglyMeasurable g
hfg : f =ᶠ[ae μ] g
⊢ ∫ (x : α), f x ∂μ ∈ s

Proof state:
case h.a
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
s : Set E
f : α → E
inst✝ : IsProbabilityMeasure μ
hs : Convex ℝ s
hsc : IsClosed s
hf : ∀ᵐ (x : α) ∂μ, f x ∈ s
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : α → E
hfi : Integrable g μ
hgm : StronglyMeasurable g
hfg : f =ᶠ[ae μ] g
this : SeparableSpace ↑(range g ∩ s)
y₀ : E
h₀ : y₀ ∈ range g ∩ s
x : α
hx : g x ∈ s
⊢ g x ∈ range g ∩ s

Tactic:
exact ⟨mem_range_self _, hx⟩

Result:
no goals

Proof state:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
s : Set E
f : α → E
inst✝ : IsProbabilityMeasure μ
hs : Convex ℝ s
hsc : IsClosed s
hf : ∀ᵐ (x : α) ∂μ, f x ∈ s
hfi : Integrable f μ
⊢ ∫ (x : α), f x ∂μ ∈ s

Tactic:
borelize E

Result:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
s : Set E
f : α → E
inst✝ : IsProbabilityMeasure μ
hs : Convex ℝ s
hsc : IsClosed s
hf : ∀ᵐ (x : α) ∂μ, f x ∈ s
hfi : Integrable f μ
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
⊢ ∫ (x : α), f x ∂μ ∈ s

Proof state:
case intro.intro.intro
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
s : Set E
f : α → E
inst✝ : IsProbabilityMeasure μ
hs : Convex ℝ s
hsc : IsClosed s
hf : ∀ᵐ (x : α) ∂μ, f x ∈ s
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : α → E
hfi : Integrable g μ
hgm : StronglyMeasurable g
hfg : f =ᶠ[ae μ] g
this : SeparableSpace ↑(range g ∩ s)
y₀ : E
h₀ : y₀ ∈ range g ∩ s
hg : ∀ᵐ (x : α) ∂μ, g x ∈ closure (range g ∩ s)
G : ℕ → SimpleFunc α E := SimpleFunc.approxOn g ⋯ (range g ∩ s) y₀ h₀
⊢ ∫ (a : α), g a ∂μ ∈ s

Tactic:
have : Tendsto (fun n => (G n).integral μ) atTop (𝓝 <| ∫ x, g x ∂μ) :=
  tendsto_integral_approxOn_of_measurable hfi _ hg _ (integrable_const _)

Result:
case intro.intro.intro
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
s : Set E
f : α → E
inst✝ : IsProbabilityMeasure μ
hs : Convex ℝ s
hsc : IsClosed s
hf : ∀ᵐ (x : α) ∂μ, f x ∈ s
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
g : α → E
hfi : Integrable g μ
hgm : StronglyMeasurable g
hfg : f =ᶠ[ae μ] g
this✝ : SeparableSpace ↑(range g ∩ s)
y₀ : E
h₀ : y₀ ∈ range g ∩ s
hg : ∀ᵐ (x : α) ∂μ, g x ∈ closure (range g ∩ s)
G : ℕ → SimpleFunc α E := SimpleFunc.approxOn g ⋯ (range g ∩ s) y₀ h₀
this : Tendsto (fun n => SimpleFunc.integral μ (G n)) atTop (𝓝 (∫ (x : α), g x ∂μ))
⊢ ∫ (a : α), g a ∂μ ∈ s

Proof state:
case intro.intro.intro
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
s : Set E
f : α → E
inst✝ : IsProbabilityMeasure μ
hs : Convex ℝ s
hsc : IsClosed s
hf : ∀ᵐ (x : α) ∂μ, f x ∈ s
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : α → E
hfi : Integrable g μ
hgm : StronglyMeasurable g
hfg : f =ᶠ[ae μ] g
this : SeparableSpace ↑(range g ∩ s)
y₀ : E
h₀ : y₀ ∈ range g ∩ s
⊢ ∫ (a : α), g a ∂μ ∈ s

Tactic:
have hg : ∀ᵐ x ∂μ, g x ∈ closure (range g ∩ s) := by
  filter_upwards [hfg.rw (fun _ y => y ∈ s) hf] with x hx
  apply subset_closure
  exact ⟨mem_range_self _, hx⟩

Result:
case intro.intro.intro
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
s : Set E
f : α → E
inst✝ : IsProbabilityMeasure μ
hs : Convex ℝ s
hsc : IsClosed s
hf : ∀ᵐ (x : α) ∂μ, f x ∈ s
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : α → E
hfi : Integrable g μ
hgm : StronglyMeasurable g
hfg : f =ᶠ[ae μ] g
this : SeparableSpace ↑(range g ∩ s)
y₀ : E
h₀ : y₀ ∈ range g ∩ s
hg : ∀ᵐ (x : α) ∂μ, g x ∈ closure (range g ∩ s)
⊢ ∫ (a : α), g a ∂μ ∈ s

Proof state:
case intro.intro.intro.refine_2
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
s : Set E
f : α → E
inst✝ : IsProbabilityMeasure μ
hs : Convex ℝ s
hsc : IsClosed s
hf : ∀ᵐ (x : α) ∂μ, f x ∈ s
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
g : α → E
hfi : Integrable g μ
hgm : StronglyMeasurable g
hfg : f =ᶠ[ae μ] g
this✝ : SeparableSpace ↑(range g ∩ s)
y₀ : E
h₀ : y₀ ∈ range g ∩ s
hg : ∀ᵐ (x : α) ∂μ, g x ∈ closure (range g ∩ s)
G : ℕ → SimpleFunc α E := SimpleFunc.approxOn g ⋯ (range g ∩ s) y₀ h₀
this : Tendsto (fun n => SimpleFunc.integral μ (G n)) atTop (𝓝 (∫ (x : α), g x ∂μ))
n : ℕ
⊢ ∀ a ∈ (G n).range, μ (⇑(G n) ⁻¹' {a}) ≠ ⊤

Tactic:
exact fun _ _ => measure_ne_top _ _

Result:
no goals

Proof state:
case intro.intro.intro.refine_2
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
s : Set E
f : α → E
inst✝ : IsProbabilityMeasure μ
hs : Convex ℝ s
hsc : IsClosed s
hf : ∀ᵐ (x : α) ∂μ, f x ∈ s
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
g : α → E
hfi : Integrable g μ
hgm : StronglyMeasurable g
hfg : f =ᶠ[ae μ] g
this✝ : SeparableSpace ↑(range g ∩ s)
y₀ : E
h₀ : y₀ ∈ range g ∩ s
hg : ∀ᵐ (x : α) ∂μ, g x ∈ closure (range g ∩ s)
G : ℕ → SimpleFunc α E := SimpleFunc.approxOn g ⋯ (range g ∩ s) y₀ h₀
this : Tendsto (fun n => SimpleFunc.integral μ (G n)) atTop (𝓝 (∫ (x : α), g x ∂μ))
n : ℕ
⊢ ∑ i ∈ (G n).range, (μ (⇑(G n) ⁻¹' {i})).toReal = 1

Tactic:
rw [← ENNReal.toReal_sum, (G n).sum_range_measure_preimage_singleton, measure_univ,
  ENNReal.one_toReal]

Result:
case intro.intro.intro.refine_2
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
s : Set E
f : α → E
inst✝ : IsProbabilityMeasure μ
hs : Convex ℝ s
hsc : IsClosed s
hf : ∀ᵐ (x : α) ∂μ, f x ∈ s
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
g : α → E
hfi : Integrable g μ
hgm : StronglyMeasurable g
hfg : f =ᶠ[ae μ] g
this✝ : SeparableSpace ↑(range g ∩ s)
y₀ : E
h₀ : y₀ ∈ range g ∩ s
hg : ∀ᵐ (x : α) ∂μ, g x ∈ closure (range g ∩ s)
G : ℕ → SimpleFunc α E := SimpleFunc.approxOn g ⋯ (range g ∩ s) y₀ h₀
this : Tendsto (fun n => SimpleFunc.integral μ (G n)) atTop (𝓝 (∫ (x : α), g x ∂μ))
n : ℕ
⊢ ∀ a ∈ (G n).range, μ (⇑(G n) ⁻¹' {a}) ≠ ⊤