Theorem: MeasureTheory.le_measure_compl_liminf_of_limsup_measure_le
File: Mathlib/MeasureTheory/Measure/Portmanteau.lean

Proof steps:
state_before: Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : limsup (fun i => (μs i) E) L ≤ μ E
⊢ μ Eᶜ ≤ liminf (fun i => (μs i) Eᶜ) L
state_after: case inl
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : limsup (fun i => (μs i) E) ⊥ ≤ μ E
⊢ μ Eᶜ ≤ liminf (fun i => (μs i) Eᶜ) ⊥

case inr
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : limsup (fun i => (μs i) E) L ≤ μ E
hne : L.NeBot
⊢ μ Eᶜ ≤ liminf (fun i => (μs i) Eᶜ) L
tactic: rcases L.eq_or_neBot with rfl | hne
state_before: case inr
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : limsup (fun i => (μs i) E) L ≤ μ E
hne : L.NeBot
⊢ μ Eᶜ ≤ liminf (fun i => (μs i) Eᶜ) L
state_after: case inr
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : limsup (fun i => (μs i) E) L ≤ μ E
hne : L.NeBot
meas_Ec : μ Eᶜ = 1 - μ E
⊢ μ Eᶜ ≤ liminf (fun i => (μs i) Eᶜ) L
tactic: have meas_Ec : μ Eᶜ = 1 - μ E := by
    simpa only [measure_univ] using measure_compl E_mble (measure_lt_top μ E).ne
state_before: case inr
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : limsup (fun i => (μs i) E) L ≤ μ E
hne : L.NeBot
meas_Ec : μ Eᶜ = 1 - μ E
⊢ μ Eᶜ ≤ liminf (fun i => (μs i) Eᶜ) L
state_after: case inr
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : limsup (fun i => (μs i) E) L ≤ μ E
hne : L.NeBot
meas_Ec : μ Eᶜ = 1 - μ E
meas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E
⊢ μ Eᶜ ≤ liminf (fun i => (μs i) Eᶜ) L
tactic: have meas_i_Ec : ∀ i, μs i Eᶜ = 1 - μs i E := by
    intro i
    simpa only [measure_univ] using measure_compl E_mble (measure_lt_top (μs i) E).ne
state_before: case inr
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : limsup (fun i => (μs i) E) L ≤ μ E
hne : L.NeBot
meas_Ec : μ Eᶜ = 1 - μ E
meas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E
⊢ μ Eᶜ ≤ liminf (fun i => (μs i) Eᶜ) L
state_after: case inr
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : limsup (fun i => (μs i) E) L ≤ μ E
hne : L.NeBot
meas_Ec : μ Eᶜ = 1 - μ E
meas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E
⊢ 1 - μ E ≤ liminf (fun i => 1 - (μs i) E) L
tactic: simp_rw [meas_Ec, meas_i_Ec]
state_before: case inr
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : limsup (fun i => (μs i) E) L ≤ μ E
hne : L.NeBot
meas_Ec : μ Eᶜ = 1 - μ E
meas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E
⊢ 1 - μ E ≤ liminf (fun i => 1 - (μs i) E) L
state_after: case inr
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : limsup (fun i => (μs i) E) L ≤ μ E
hne : L.NeBot
meas_Ec : μ Eᶜ = 1 - μ E
meas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E
obs : liminf (fun i => 1 - (μs i) E) L = liminf ((fun x => 1 - x) ∘ fun i => (μs i) E) L
⊢ 1 - μ E ≤ liminf (fun i => 1 - (μs i) E) L
tactic: have obs :
    (L.liminf fun i : ι => 1 - μs i E) = L.liminf ((fun x => 1 - x) ∘ fun i : ι => μs i E) := rfl
state_before: case inr
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : limsup (fun i => (μs i) E) L ≤ μ E
hne : L.NeBot
meas_Ec : μ Eᶜ = 1 - μ E
meas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E
obs : liminf (fun i => 1 - (μs i) E) L = liminf ((fun x => 1 - x) ∘ fun i => (μs i) E) L
⊢ 1 - μ E ≤ liminf (fun i => 1 - (μs i) E) L
state_after: case inr
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : limsup (fun i => (μs i) E) L ≤ μ E
hne : L.NeBot
meas_Ec : μ Eᶜ = 1 - μ E
meas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E
obs : liminf (fun i => 1 - (μs i) E) L = liminf ((fun x => 1 - x) ∘ fun i => (μs i) E) L
⊢ 1 - μ E ≤ liminf ((fun x => 1 - x) ∘ fun i => (μs i) E) L
tactic: rw [obs]
state_before: case inr
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : limsup (fun i => (μs i) E) L ≤ μ E
hne : L.NeBot
meas_Ec : μ Eᶜ = 1 - μ E
meas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E
obs : liminf (fun i => 1 - (μs i) E) L = liminf ((fun x => 1 - x) ∘ fun i => (μs i) E) L
⊢ 1 - μ E ≤ liminf ((fun x => 1 - x) ∘ fun i => (μs i) E) L
state_after: case inr
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : limsup (fun i => (μs i) E) L ≤ μ E
hne : L.NeBot
meas_Ec : μ Eᶜ = 1 - μ E
meas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E
obs : liminf (fun i => 1 - (μs i) E) L = liminf ((fun x => 1 - x) ∘ fun i => (μs i) E) L
this : 1 - limsup (fun i => (μs i) E) L = liminf ((fun x => 1 - x) ∘ fun i => (μs i) E) L
⊢ 1 - μ E ≤ liminf ((fun x => 1 - x) ∘ fun i => (μs i) E) L
tactic: have := antitone_const_tsub.map_limsup_of_continuousAt (F := L)
    (fun i => μs i E) (ENNReal.continuous_sub_left ENNReal.one_ne_top).continuousAt
state_before: case inr
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : limsup (fun i => (μs i) E) L ≤ μ E
hne : L.NeBot
meas_Ec : μ Eᶜ = 1 - μ E
meas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E
obs : liminf (fun i => 1 - (μs i) E) L = liminf ((fun x => 1 - x) ∘ fun i => (μs i) E) L
this : 1 - limsup (fun i => (μs i) E) L = liminf ((fun x => 1 - x) ∘ fun i => (μs i) E) L
⊢ 1 - μ E ≤ liminf ((fun x => 1 - x) ∘ fun i => (μs i) E) L
state_after: case inr
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : limsup (fun i => (μs i) E) L ≤ μ E
hne : L.NeBot
meas_Ec : μ Eᶜ = 1 - μ E
meas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E
obs : liminf (fun i => 1 - (μs i) E) L = liminf ((fun x => 1 - x) ∘ fun i => (μs i) E) L
this : 1 - limsup (fun i => (μs i) E) L = liminf ((fun x => 1 - x) ∘ fun i => (μs i) E) L
⊢ 1 - μ E ≤ 1 - limsup (fun i => (μs i) E) L
tactic: simp_rw [← this]
state_before: case inr
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : limsup (fun i => (μs i) E) L ≤ μ E
hne : L.NeBot
meas_Ec : μ Eᶜ = 1 - μ E
meas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E
obs : liminf (fun i => 1 - (μs i) E) L = liminf ((fun x => 1 - x) ∘ fun i => (μs i) E) L
this : 1 - limsup (fun i => (μs i) E) L = liminf ((fun x => 1 - x) ∘ fun i => (μs i) E) L
⊢ 1 - μ E ≤ 1 - limsup (fun i => (μs i) E) L
state_after: no goals
tactic: exact antitone_const_tsub h