Theorem: MeasureTheory.pdf.IsUniform.integral_eq
File: Mathlib/Probability/Distributions/Uniform.lean

tactic: rw [← smul_eq_mul, ← integral_smul_measure]
state_before: E : Type u_1
inst✝ : MeasurableSpace E
m μ : Measure E
Ω : Type u_2
x✝ : MeasurableSpace Ω
ℙ : Measure Ω
X : Ω → ℝ
s : Set ℝ
huX : IsUniform X s ℙ volume
⊢ ∫ (x : Ω), X x ∂ℙ = (volume s)⁻¹.toReal * ∫ (x : ℝ) in s, x
state_after: E : Type u_1
inst✝ : MeasurableSpace E
m μ : Measure E
Ω : Type u_2
x✝ : MeasurableSpace Ω
ℙ : Measure Ω
X : Ω → ℝ
s : Set ℝ
huX : IsUniform X s ℙ volume
⊢ ∫ (x : Ω), X x ∂ℙ = ∫ (x : ℝ), x ∂(volume s)⁻¹ • volume.restrict s

tactic: dsimp only [IsUniform, ProbabilityTheory.cond] at huX
state_before: E : Type u_1
inst✝ : MeasurableSpace E
m μ : Measure E
Ω : Type u_2
x✝ : MeasurableSpace Ω
ℙ : Measure Ω
X : Ω → ℝ
s : Set ℝ
huX : IsUniform X s ℙ volume
⊢ ∫ (x : Ω), X x ∂ℙ = ∫ (x : ℝ), x ∂(volume s)⁻¹ • volume.restrict s
state_after: E : Type u_1
inst✝ : MeasurableSpace E
m μ : Measure E
Ω : Type u_2
x✝ : MeasurableSpace Ω
ℙ : Measure Ω
X : Ω → ℝ
s : Set ℝ
huX : Measure.map X ℙ = (volume s)⁻¹ • volume.restrict s
⊢ ∫ (x : Ω), X x ∂ℙ = ∫ (x : ℝ), x ∂(volume s)⁻¹ • volume.restrict s

tactic: rw [← huX]
state_before: E : Type u_1
inst✝ : MeasurableSpace E
m μ : Measure E
Ω : Type u_2
x✝ : MeasurableSpace Ω
ℙ : Measure Ω
X : Ω → ℝ
s : Set ℝ
huX : Measure.map X ℙ = (volume s)⁻¹ • volume.restrict s
⊢ ∫ (x : Ω), X x ∂ℙ = ∫ (x : ℝ), x ∂(volume s)⁻¹ • volume.restrict s
state_after: E : Type u_1
inst✝ : MeasurableSpace E
m μ : Measure E
Ω : Type u_2
x✝ : MeasurableSpace Ω
ℙ : Measure Ω
X : Ω → ℝ
s : Set ℝ
huX : Measure.map X ℙ = (volume s)⁻¹ • volume.restrict s
⊢ ∫ (x : Ω), X x ∂ℙ = ∫ (x : ℝ), x ∂Measure.map X ℙ

tactic: by_cases hX : AEMeasurable X ℙ
state_before: E : Type u_1
inst✝ : MeasurableSpace E
m μ : Measure E
Ω : Type u_2
x✝ : MeasurableSpace Ω
ℙ : Measure Ω
X : Ω → ℝ
s : Set ℝ
huX : Measure.map X ℙ = (volume s)⁻¹ • volume.restrict s
⊢ ∫ (x : Ω), X x ∂ℙ = ∫ (x : ℝ), x ∂Measure.map X ℙ
state_after: case pos
E : Type u_1
inst✝ : MeasurableSpace E
m μ : Measure E
Ω : Type u_2
x✝ : MeasurableSpace Ω
ℙ : Measure Ω
X : Ω → ℝ
s : Set ℝ
huX : Measure.map X ℙ = (volume s)⁻¹ • volume.restrict s
hX : AEMeasurable X ℙ
⊢ ∫ (x : Ω), X x ∂ℙ = ∫ (x : ℝ), x ∂Measure.map X ℙ

case neg
E : Type u_1
inst✝ : MeasurableSpace E
m μ : Measure E
Ω : Type u_2
x✝ : MeasurableSpace Ω
ℙ : Measure Ω
X : Ω → ℝ
s : Set ℝ
huX : Measure.map X ℙ = (volume s)⁻¹ • volume.restrict s
hX : ¬AEMeasurable X ℙ
⊢ ∫ (x : Ω), X x ∂ℙ = ∫ (x : ℝ), x ∂Measure.map X ℙ

tactic: exact (integral_map hX aestronglyMeasurable_id).symm
state_before: case pos
E : Type u_1
inst✝ : MeasurableSpace E
m μ : Measure E
Ω : Type u_2
x✝ : MeasurableSpace Ω
ℙ : Measure Ω
X : Ω → ℝ
s : Set ℝ
huX : Measure.map X ℙ = (volume s)⁻¹ • volume.restrict s
hX : AEMeasurable X ℙ
⊢ ∫ (x : Ω), X x ∂ℙ = ∫ (x : ℝ), x ∂Measure.map X ℙ
state_after: no goals

tactic: rw [map_of_not_aemeasurable hX, integral_zero_measure, integral_non_aestronglyMeasurable]
state_before: case neg
E : Type u_1
inst✝ : MeasurableSpace E
m μ : Measure E
Ω : Type u_2
x✝ : MeasurableSpace Ω
ℙ : Measure Ω
X : Ω → ℝ
s : Set ℝ
huX : Measure.map X ℙ = (volume s)⁻¹ • volume.restrict s
hX : ¬AEMeasurable X ℙ
⊢ ∫ (x : Ω), X x ∂ℙ = ∫ (x : ℝ), x ∂Measure.map X ℙ
state_after: case neg
E : Type u_1
inst✝ : MeasurableSpace E
m μ : Measure E
Ω : Type u_2
x✝ : MeasurableSpace Ω
ℙ : Measure Ω
X : Ω → ℝ
s : Set ℝ
huX : Measure.map X ℙ = (volume s)⁻¹ • volume.restrict s
hX : ¬AEMeasurable X ℙ
⊢ ¬AEStronglyMeasurable X ℙ

tactic: rwa [aestronglyMeasurable_iff_aemeasurable]
state_before: case neg
E : Type u_1
inst✝ : MeasurableSpace E
m μ : Measure E
Ω : Type u_2
x✝ : MeasurableSpace Ω
ℙ : Measure Ω
X : Ω → ℝ
s : Set ℝ
huX : Measure.map X ℙ = (volume s)⁻¹ • volume.restrict s
hX : ¬AEMeasurable X ℙ
⊢ ¬AEStronglyMeasurable X ℙ
state_after: no goals