Theorem: BoundedContinuousFunction.tendsto_integral_of_forall_integral_le_liminf_integral
File: Mathlib/MeasureTheory/Integral/BoundedContinuousFunction.lean

Proof steps:
state_before: X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace X
inst✝² : OpensMeasurableSpace X
ι : Type u_2
L : Filter ι
μ : Measure X
inst✝¹ : IsProbabilityMeasure μ
μs : ι → Measure X
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
h : ∀ (f : X →ᵇ ℝ), 0 ≤ f → ∫ (x : X), f x ∂μ ≤ liminf (fun i => ∫ (x : X), f x ∂μs i) L
f : X →ᵇ ℝ
⊢ Tendsto (fun i => ∫ (x : X), f x ∂μs i) L (𝓝 (∫ (x : X), f x ∂μ))
state_after: case inl
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace X
inst✝² : OpensMeasurableSpace X
ι : Type u_2
μ : Measure X
inst✝¹ : IsProbabilityMeasure μ
μs : ι → Measure X
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
f : X →ᵇ ℝ
h : ∀ (f : X →ᵇ ℝ), 0 ≤ f → ∫ (x : X), f x ∂μ ≤ liminf (fun i => ∫ (x : X), f x ∂μs i) ⊥
⊢ Tendsto (fun i => ∫ (x : X), f x ∂μs i) ⊥ (𝓝 (∫ (x : X), f x ∂μ))

case inr
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace X
inst✝² : OpensMeasurableSpace X
ι : Type u_2
L : Filter ι
μ : Measure X
inst✝¹ : IsProbabilityMeasure μ
μs : ι → Measure X
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
h : ∀ (f : X →ᵇ ℝ), 0 ≤ f → ∫ (x : X), f x ∂μ ≤ liminf (fun i => ∫ (x : X), f x ∂μs i) L
f : X →ᵇ ℝ
hL : L.NeBot
⊢ Tendsto (fun i => ∫ (x : X), f x ∂μs i) L (𝓝 (∫ (x : X), f x ∂μ))
tactic: rcases eq_or_neBot L with rfl|hL
state_before: case inr
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace X
inst✝² : OpensMeasurableSpace X
ι : Type u_2
L : Filter ι
μ : Measure X
inst✝¹ : IsProbabilityMeasure μ
μs : ι → Measure X
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
h : ∀ (f : X →ᵇ ℝ), 0 ≤ f → ∫ (x : X), f x ∂μ ≤ liminf (fun i => ∫ (x : X), f x ∂μs i) L
f : X →ᵇ ℝ
hL : L.NeBot
⊢ Tendsto (fun i => ∫ (x : X), f x ∂μs i) L (𝓝 (∫ (x : X), f x ∂μ))
state_after: case inr
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace X
inst✝² : OpensMeasurableSpace X
ι : Type u_2
L : Filter ι
μ : Measure X
inst✝¹ : IsProbabilityMeasure μ
μs : ι → Measure X
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
h : ∀ (f : X →ᵇ ℝ), 0 ≤ f → ∫ (x : X), f x ∂μ ≤ liminf (fun i => ∫ (x : X), f x ∂μs i) L
f : X →ᵇ ℝ
hL : L.NeBot
obs : Bornology.IsBounded (Set.range fun i => ∫ (x : X), f x ∂μs i)
⊢ Tendsto (fun i => ∫ (x : X), f x ∂μs i) L (𝓝 (∫ (x : X), f x ∂μ))
tactic: have obs := BoundedContinuousFunction.isBounded_range_integral μs f
state_before: case inr
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace X
inst✝² : OpensMeasurableSpace X
ι : Type u_2
L : Filter ι
μ : Measure X
inst✝¹ : IsProbabilityMeasure μ
μs : ι → Measure X
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
h : ∀ (f : X →ᵇ ℝ), 0 ≤ f → ∫ (x : X), f x ∂μ ≤ liminf (fun i => ∫ (x : X), f x ∂μs i) L
f : X →ᵇ ℝ
hL : L.NeBot
obs : Bornology.IsBounded (Set.range fun i => ∫ (x : X), f x ∂μs i)
bdd_above : IsBoundedUnder (fun x x_1 => x ≤ x_1) L fun i => ∫ (x : X), f x ∂μs i
bdd_below : IsBoundedUnder (fun x x_1 => x ≥ x_1) L fun i => ∫ (x : X), f x ∂μs i
⊢ Tendsto (fun i => ∫ (x : X), f x ∂μs i) L (𝓝 (∫ (x : X), f x ∂μ))
state_after: case inr.hinf
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace X
inst✝² : OpensMeasurableSpace X
ι : Type u_2
L : Filter ι
μ : Measure X
inst✝¹ : IsProbabilityMeasure μ
μs : ι → Measure X
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
h : ∀ (f : X →ᵇ ℝ), 0 ≤ f → ∫ (x : X), f x ∂μ ≤ liminf (fun i => ∫ (x : X), f x ∂μs i) L
f : X →ᵇ ℝ
hL : L.NeBot
obs : Bornology.IsBounded (Set.range fun i => ∫ (x : X), f x ∂μs i)
bdd_above : IsBoundedUnder (fun x x_1 => x ≤ x_1) L fun i => ∫ (x : X), f x ∂μs i
bdd_below : IsBoundedUnder (fun x x_1 => x ≥ x_1) L fun i => ∫ (x : X), f x ∂μs i
⊢ ∫ (x : X), f x ∂μ ≤ liminf (fun i => ∫ (x : X), f x ∂μs i) L

case inr.hsup
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace X
inst✝² : OpensMeasurableSpace X
ι : Type u_2
L : Filter ι
μ : Measure X
inst✝¹ : IsProbabilityMeasure μ
μs : ι → Measure X
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
h : ∀ (f : X →ᵇ ℝ), 0 ≤ f → ∫ (x : X), f x ∂μ ≤ liminf (fun i => ∫ (x : X), f x ∂μs i) L
f : X →ᵇ ℝ
hL : L.NeBot
obs : Bornology.IsBounded (Set.range fun i => ∫ (x : X), f x ∂μs i)
bdd_above : IsBoundedUnder (fun x x_1 => x ≤ x_1) L fun i => ∫ (x : X), f x ∂μs i
bdd_below : IsBoundedUnder (fun x x_1 => x ≥ x_1) L fun i => ∫ (x : X), f x ∂μs i
⊢ limsup (fun i => ∫ (x : X), f x ∂μs i) L ≤ ∫ (x : X), f x ∂μ

case inr.h
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace X
inst✝² : OpensMeasurableSpace X
ι : Type u_2
L : Filter ι
μ : Measure X
inst✝¹ : IsProbabilityMeasure μ
μs : ι → Measure X
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
h : ∀ (f : X →ᵇ ℝ), 0 ≤ f → ∫ (x : X), f x ∂μ ≤ liminf (fun i => ∫ (x : X), f x ∂μs i) L
f : X →ᵇ ℝ
hL : L.NeBot
obs : Bornology.IsBounded (Set.range fun i => ∫ (x : X), f x ∂μs i)
bdd_above : IsBoundedUnder (fun x x_1 => x ≤ x_1) L fun i => ∫ (x : X), f x ∂μs i
bdd_below : IsBoundedUnder (fun x x_1 => x ≥ x_1) L fun i => ∫ (x : X), f x ∂μs i
⊢ autoParam (IsBoundedUnder (fun x x_1 => x ≤ x_1) L fun i => ∫ (x : X), f x ∂μs i) _auto✝

case inr.h'
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace X
inst✝² : OpensMeasurableSpace X
ι : Type u_2
L : Filter ι
μ : Measure X
inst✝¹ : IsProbabilityMeasure μ
μs : ι → Measure X
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
h : ∀ (f : X →ᵇ ℝ), 0 ≤ f → ∫ (x : X), f x ∂μ ≤ liminf (fun i => ∫ (x : X), f x ∂μs i) L
f : X →ᵇ ℝ
hL : L.NeBot
obs : Bornology.IsBounded (Set.range fun i => ∫ (x : X), f x ∂μs i)
bdd_above : IsBoundedUnder (fun x x_1 => x ≤ x_1) L fun i => ∫ (x : X), f x ∂μs i
bdd_below : IsBoundedUnder (fun x x_1 => x ≥ x_1) L fun i => ∫ (x : X), f x ∂μs i
⊢ autoParam (IsBoundedUnder (fun x x_1 => x ≥ x_1) L fun i => ∫ (x : X), f x ∂μs i) _auto✝
tactic: apply @tendsto_of_le_liminf_of_limsup_le ℝ ι _ _ _ L (fun i ↦ ∫ x, f x ∂ (μs i)) (∫ x, f x ∂μ)