Theorem: MeasureTheory.BoundedContinuousFunction.integral_le_of_levyProkhorovEDist_lt
File: Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean

Proof steps:
state_before: ι : Type u_1
Ω : Type u_2
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
ε : ℝ
ε_pos : 0 < ε
hμν : levyProkhorovEDist μ ν < ENNReal.ofReal ε
f : Ω →ᵇ ℝ
f_nn : 0 ≤ᶠ[ae μ] ⇑f
⊢ ∫ (ω : Ω), f ω ∂μ ≤ (∫ (t : ℝ) in Ioc 0 ‖f‖, (ν (thickening ε {a | t ≤ f a})).toReal) + ε * ‖f‖
state_after: ι : Type u_1
Ω : Type u_2
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
ε : ℝ
ε_pos : 0 < ε
hμν : levyProkhorovEDist μ ν < ENNReal.ofReal ε
f : Ω →ᵇ ℝ
f_nn : 0 ≤ᶠ[ae μ] ⇑f
⊢ ∫ (t : ℝ) in Ioc 0 ‖f‖, (μ {a | t ≤ f a}).toReal ≤
    (∫ (t : ℝ) in Ioc 0 ‖f‖, (ν (thickening ε {a | t ≤ f a})).toReal) + ε * ‖f‖
tactic: rw [BoundedContinuousFunction.integral_eq_integral_meas_le f μ f_nn]
state_before: ι : Type u_1
Ω : Type u_2
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
ε : ℝ
ε_pos : 0 < ε
hμν : levyProkhorovEDist μ ν < ENNReal.ofReal ε
f : Ω →ᵇ ℝ
f_nn : 0 ≤ᶠ[ae μ] ⇑f
key : (fun t => (μ {a | t ≤ f a}).toReal) ≤ fun t => (ν (thickening ε {a | t ≤ f a})).toReal + ε
intble₁ : IntegrableOn (fun t => (μ {a | t ≤ f a}).toReal) (Ioc 0 ‖f‖) volume
intble₂ : IntegrableOn (fun t => (ν (thickening ε {a | t ≤ f a})).toReal) (Ioc 0 ‖f‖) volume
⊢ ∫ (t : ℝ) in Ioc 0 ‖f‖, (μ {a | t ≤ f a}).toReal ≤
    (∫ (t : ℝ) in Ioc 0 ‖f‖, (ν (thickening ε {a | t ≤ f a})).toReal) + ε * ‖f‖
state_after: ι : Type u_1
Ω : Type u_2
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
ε : ℝ
ε_pos : 0 < ε
hμν : levyProkhorovEDist μ ν < ENNReal.ofReal ε
f : Ω →ᵇ ℝ
f_nn : 0 ≤ᶠ[ae μ] ⇑f
key : (fun t => (μ {a | t ≤ f a}).toReal) ≤ fun t => (ν (thickening ε {a | t ≤ f a})).toReal + ε
intble₁ : IntegrableOn (fun t => (μ {a | t ≤ f a}).toReal) (Ioc 0 ‖f‖) volume
intble₂ : IntegrableOn (fun t => (ν (thickening ε {a | t ≤ f a})).toReal) (Ioc 0 ‖f‖) volume
⊢ ∫ (x : ℝ) in Ioc 0 ‖f‖, (ν (thickening ε {a | x ≤ f a})).toReal + ε ∂volume ≤
    (∫ (t : ℝ) in Ioc 0 ‖f‖, (ν (thickening ε {a | t ≤ f a})).toReal) + ε * ‖f‖

ι : Type u_1
Ω : Type u_2
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
ε : ℝ
ε_pos : 0 < ε
hμν : levyProkhorovEDist μ ν < ENNReal.ofReal ε
f : Ω →ᵇ ℝ
f_nn : 0 ≤ᶠ[ae μ] ⇑f
key : (fun t => (μ {a | t ≤ f a}).toReal) ≤ fun t => (ν (thickening ε {a | t ≤ f a})).toReal + ε
intble₁ : IntegrableOn (fun t => (μ {a | t ≤ f a}).toReal) (Ioc 0 ‖f‖) volume
intble₂ : IntegrableOn (fun t => (ν (thickening ε {a | t ≤ f a})).toReal) (Ioc 0 ‖f‖) volume
⊢ IntegrableOn (fun t => (μ {a | t ≤ f a}).toReal) (Ioc 0 ‖f‖) volume

ι : Type u_1
Ω : Type u_2
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
ε : ℝ
ε_pos : 0 < ε
hμν : levyProkhorovEDist μ ν < ENNReal.ofReal ε
f : Ω →ᵇ ℝ
f_nn : 0 ≤ᶠ[ae μ] ⇑f
key : (fun t => (μ {a | t ≤ f a}).toReal) ≤ fun t => (ν (thickening ε {a | t ≤ f a})).toReal + ε
intble₁ : IntegrableOn (fun t => (μ {a | t ≤ f a}).toReal) (Ioc 0 ‖f‖) volume
intble₂ : IntegrableOn (fun t => (ν (thickening ε {a | t ≤ f a})).toReal) (Ioc 0 ‖f‖) volume
⊢ IntegrableOn (fun t => (ν (thickening ε {a | t ≤ f a})).toReal + ε) (Ioc 0 ‖f‖) volume
tactic: apply le_trans (setIntegral_mono (s := Ioc 0 ‖f‖) ?_ ?_ key)