Theorem: MeasureTheory.Measure.QuasiMeasurePreserving.liminf_preimage_iterate_ae_eq
File: Mathlib/MeasureTheory/Measure/MeasureSpace.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
f✝¹ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f✝ : α → β
f : α → α
hf : QuasiMeasurePreserving f μ μ
hs : f ⁻¹' s =ᵐ[μ] s
⊢ liminf (fun n => (preimage f)^[n] s) atTop =ᵐ[μ] s
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
f✝¹ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f✝ : α → β
f : α → α
hf : QuasiMeasurePreserving f μ μ
hs : f ⁻¹' s =ᵐ[μ] s
⊢ limsup (compl ∘ fun n => (preimage f)^[n] s) atTop =ᵐ[μ] sᶜ
tactic: rw [← ae_eq_set_compl_compl, @Filter.liminf_compl (Set α)]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
f✝¹ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f✝ : α → β
f : α → α
hf : QuasiMeasurePreserving f μ μ
hs : f ⁻¹' s =ᵐ[μ] s
⊢ limsup (compl ∘ fun n => (preimage f)^[n] s) atTop =ᵐ[μ] sᶜ
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
f✝¹ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f✝ : α → β
f : α → α
hf : QuasiMeasurePreserving f μ μ
hs : f ⁻¹' sᶜ =ᵐ[μ] sᶜ
⊢ limsup (compl ∘ fun n => (preimage f)^[n] s) atTop =ᵐ[μ] sᶜ
tactic: rw [← ae_eq_set_compl_compl, ← preimage_compl] at hs
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
f✝¹ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f✝ : α → β
f : α → α
hf : QuasiMeasurePreserving f μ μ
hs : f ⁻¹' sᶜ =ᵐ[μ] sᶜ
⊢ limsup (compl ∘ fun n => (preimage f)^[n] s) atTop =ᵐ[μ] sᶜ
state_after: case h.e'_4.h.e'_4.h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
f✝¹ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f✝ : α → β
f : α → α
hf : QuasiMeasurePreserving f μ μ
hs : f ⁻¹' sᶜ =ᵐ[μ] sᶜ
x✝ : ℕ
⊢ (compl ∘ fun n => (preimage f)^[n] s) x✝ = (preimage f)^[x✝] sᶜ
tactic: convert hf.limsup_preimage_iterate_ae_eq hs
state_before: case h.e'_4.h.e'_4.h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
f✝¹ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f✝ : α → β
f : α → α
hf : QuasiMeasurePreserving f μ μ
hs : f ⁻¹' sᶜ =ᵐ[μ] sᶜ
x✝ : ℕ
⊢ (compl ∘ fun n => (preimage f)^[n] s) x✝ = (preimage f)^[x✝] sᶜ
state_after: case h.e'_4.h.e'_4.h.h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
f✝¹ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f✝ : α → β
f : α → α
hf : QuasiMeasurePreserving f μ μ
hs : f ⁻¹' sᶜ =ᵐ[μ] sᶜ
x✝ : ℕ
n : α
⊢ n ∈ (compl ∘ fun n => (preimage f)^[n] s) x✝ ↔ n ∈ (preimage f)^[x✝] sᶜ
tactic: ext1 n
state_before: case h.e'_4.h.e'_4.h.h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
f✝¹ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f✝ : α → β
f : α → α
hf : QuasiMeasurePreserving f μ μ
hs : f ⁻¹' sᶜ =ᵐ[μ] sᶜ
x✝ : ℕ
n : α
⊢ n ∈ (compl ∘ fun n => (preimage f)^[n] s) x✝ ↔ n ∈ (preimage f)^[x✝] sᶜ
state_after: no goals
tactic: simp only [← Set.preimage_iterate_eq, comp_apply, preimage_compl]