Theorem: Convex.average_mem_interior_of_set
File: Mathlib/Analysis/Convex/Integral.lean

tactic: rw [← measure_toMeasurable] at h0
state_before: α : Type u_1
E : Type u_2
F : Type ?u.1945481
m0 : MeasurableSpace α
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : CompleteSpace F
μ : MeasureTheory.Measure α
s : Set E
t : Set α
f : α → E
g : E → ℝ
C : ℝ
inst✝ : IsFiniteMeasure μ
hs : Convex ℝ s
h0 : ↑↑μ t ≠ 0
hfs : ∀ᵐ (x : α) ∂μ, f x ∈ s
hfi : Integrable f
ht : (⨍ (x : α) in t, f x ∂μ) ∈ interior s
⊢ (⨍ (x : α), f x ∂μ) ∈ interior s

tactic: rw [← restrict_toMeasurable (measure_ne_top μ t)] at ht
state_before: α : Type u_1
E : Type u_2
F : Type ?u.1945481
m0 : MeasurableSpace α
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : CompleteSpace F
μ : MeasureTheory.Measure α
s : Set E
t : Set α
f : α → E
g : E → ℝ
C : ℝ
inst✝ : IsFiniteMeasure μ
hs : Convex ℝ s
h0 : ↑↑μ (toMeasurable μ t) ≠ 0
hfs : ∀ᵐ (x : α) ∂μ, f x ∈ s
hfi : Integrable f
ht : (⨍ (x : α) in t, f x ∂μ) ∈ interior s
⊢ (⨍ (x : α), f x ∂μ) ∈ interior s

tactic: by_cases h0' : μ (toMeasurable μ tᶜ) = 0
state_before: α : Type u_1
E : Type u_2
F : Type ?u.1945481
m0 : MeasurableSpace α
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : CompleteSpace F
μ : MeasureTheory.Measure α
s : Set E
t : Set α
f : α → E
g : E → ℝ
C : ℝ
inst✝ : IsFiniteMeasure μ
hs : Convex ℝ s
h0 : ↑↑μ (toMeasurable μ t) ≠ 0
hfs : ∀ᵐ (x : α) ∂μ, f x ∈ s
hfi : Integrable f
ht : (⨍ (x : α) in toMeasurable μ t, f x ∂μ) ∈ interior s
⊢ (⨍ (x : α), f x ∂μ) ∈ interior s

tactic: exact
  hs.openSegment_interior_closure_subset_interior ht
    (hs.set_average_mem_closure h0' (measure_ne_top _ _) (ae_restrict_of_ae hfs)
      hfi.integrableOn)
    (average_mem_openSegment_compl_self (measurableSet_toMeasurable μ t).nullMeasurableSet h0
      h0' hfi)
state_before: case neg
α : Type u_1
E : Type u_2
F : Type ?u.1945481
m0 : MeasurableSpace α
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : CompleteSpace F
μ : MeasureTheory.Measure α
s : Set E
t : Set α
f : α → E
g : E → ℝ
C : ℝ
inst✝ : IsFiniteMeasure μ
hs : Convex ℝ s
h0 : ↑↑μ (toMeasurable μ t) ≠ 0
hfs : ∀ᵐ (x : α) ∂μ, f x ∈ s
hfi : Integrable f
ht : (⨍ (x : α) in toMeasurable μ t, f x ∂μ) ∈ interior s
h0' : ¬↑↑μ (toMeasurable μ tᶜ) = 0
⊢ (⨍ (x : α), f x ∂μ) ∈ interior s

tactic: rw [← ae_eq_univ] at h0'
state_before: case pos
α : Type u_1
E : Type u_2
F : Type ?u.1945481
m0 : MeasurableSpace α
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : CompleteSpace F
μ : MeasureTheory.Measure α
s : Set E
t : Set α
f : α → E
g : E → ℝ
C : ℝ
inst✝ : IsFiniteMeasure μ
hs : Convex ℝ s
h0 : ↑↑μ (toMeasurable μ t) ≠ 0
hfs : ∀ᵐ (x : α) ∂μ, f x ∈ s
hfi : Integrable f
ht : (⨍ (x : α) in toMeasurable μ t, f x ∂μ) ∈ interior s
h0' : ↑↑μ (toMeasurable μ tᶜ) = 0
⊢ (⨍ (x : α), f x ∂μ) ∈ interior s

tactic: rwa [restrict_congr_set h0', restrict_univ] at ht
state_before: case pos
α : Type u_1
E : Type u_2
F : Type ?u.1945481
m0 : MeasurableSpace α
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : CompleteSpace F
μ : MeasureTheory.Measure α
s : Set E
t : Set α
f : α → E
g : E → ℝ
C : ℝ
inst✝ : IsFiniteMeasure μ
hs : Convex ℝ s
h0 : ↑↑μ (toMeasurable μ t) ≠ 0
hfs : ∀ᵐ (x : α) ∂μ, f x ∈ s
hfi : Integrable f
ht : (⨍ (x : α) in toMeasurable μ t, f x ∂μ) ∈ interior s
h0' : toMeasurable μ t =ᶠ[ae μ] univ
⊢ (⨍ (x : α), f x ∂μ) ∈ interior s