Theorem: MeasureTheory.NullMeasurable.measure_preimage_eq_measure_restrict_preimage_of_ae_compl_eq_const
File: Mathlib/MeasureTheory/Measure/Restrict.lean

tactic: rw [Measure.restrict_apply₀ (f_mble t_mble)]
state_before: R : Type u_1
α : Type u_2
β✝ : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β✝
inst✝¹ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t✝ : Set α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : f =ᶠ[ae (μ.restrict sᶜ)] fun x => b
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
⊢ μ (f ⁻¹' t) = (μ.restrict s) (f ⁻¹' t)
state_after: R : Type u_1
α : Type u_2
β✝ : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β✝
inst✝¹ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t✝ : Set α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : f =ᶠ[ae (μ.restrict sᶜ)] fun x => b
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
⊢ μ (f ⁻¹' t) = μ (f ⁻¹' t ∩ s)

tactic: rw [EventuallyEq, ae_iff, Measure.restrict_apply₀] at hs
state_before: R : Type u_1
α : Type u_2
β✝ : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β✝
inst✝¹ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t✝ : Set α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : f =ᶠ[ae (μ.restrict sᶜ)] fun x => b
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
⊢ μ (f ⁻¹' t) = μ (f ⁻¹' t ∩ s)
state_after: R : Type u_1
α : Type u_2
β✝ : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β✝
inst✝¹ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t✝ : Set α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : μ ({a | ¬f a = b} ∩ sᶜ) = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
⊢ μ (f ⁻¹' t) = μ (f ⁻¹' t ∩ s)

R : Type u_1
α : Type u_2
β✝ : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β✝
inst✝¹ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t✝ : Set α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : (μ.restrict sᶜ) {a | ¬f a = b} = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
⊢ NullMeasurableSet {a | ¬f a = b} (μ.restrict sᶜ)

tactic: apply le_antisymm _ (measure_mono inter_subset_left)
state_before: R : Type u_1
α : Type u_2
β✝ : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β✝
inst✝¹ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t✝ : Set α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : μ ({a | ¬f a = b} ∩ sᶜ) = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
⊢ μ (f ⁻¹' t) = μ (f ⁻¹' t ∩ s)
state_after: R : Type u_1
α : Type u_2
β✝ : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β✝
inst✝¹ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t✝ : Set α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : μ ({a | ¬f a = b} ∩ sᶜ) = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
⊢ μ (f ⁻¹' t) ≤ μ (f ⁻¹' t ∩ s)

tactic: apply (measure_mono (Eq.symm (inter_union_compl (f ⁻¹' t) s)).le).trans
state_before: R : Type u_1
α : Type u_2
β✝ : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β✝
inst✝¹ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t✝ : Set α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : μ ({a | ¬f a = b} ∩ sᶜ) = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
⊢ μ (f ⁻¹' t) ≤ μ (f ⁻¹' t ∩ s)
state_after: R : Type u_1
α : Type u_2
β✝ : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β✝
inst✝¹ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t✝ : Set α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : μ ({a | ¬f a = b} ∩ sᶜ) = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
⊢ μ (f ⁻¹' t ∩ s ∪ f ⁻¹' t ∩ sᶜ) ≤ μ (f ⁻¹' t ∩ s)

tactic: apply (measure_union_le _ _).trans
state_before: R : Type u_1
α : Type u_2
β✝ : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β✝
inst✝¹ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t✝ : Set α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : μ ({a | ¬f a = b} ∩ sᶜ) = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
⊢ μ (f ⁻¹' t ∩ s ∪ f ⁻¹' t ∩ sᶜ) ≤ μ (f ⁻¹' t ∩ s)
state_after: R : Type u_1
α : Type u_2
β✝ : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β✝
inst✝¹ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t✝ : Set α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : μ ({a | ¬f a = b} ∩ sᶜ) = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
⊢ μ (f ⁻¹' t ∩ s) + μ (f ⁻¹' t ∩ sᶜ) ≤ μ (f ⁻¹' t ∩ s)

tactic: have obs : μ ((f ⁻¹' t) ∩ sᶜ) = 0 := by
  apply le_antisymm _ (zero_le _)
  rw [← hs]
  apply measure_mono (inter_subset_inter_left _ _)
  intro x hx hfx
  simp only [mem_preimage, mem_setOf_eq] at hx hfx
  exact ht (hfx ▸ hx)
state_before: R : Type u_1
α : Type u_2
β✝ : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β✝
inst✝¹ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t✝ : Set α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : μ ({a | ¬f a = b} ∩ sᶜ) = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
⊢ μ (f ⁻¹' t ∩ s) + μ (f ⁻¹' t ∩ sᶜ) ≤ μ (f ⁻¹' t ∩ s)
state_after: R : Type u_1
α : Type u_2
β✝ : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β✝
inst✝¹ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t✝ : Set α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : μ ({a | ¬f a = b} ∩ sᶜ) = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
obs : μ (f ⁻¹' t ∩ sᶜ) = 0
⊢ μ (f ⁻¹' t ∩ s) + μ (f ⁻¹' t ∩ sᶜ) ≤ μ (f ⁻¹' t ∩ s)

tactic: simp only [obs, add_zero, le_refl]
state_before: R : Type u_1
α : Type u_2
β✝ : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β✝
inst✝¹ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t✝ : Set α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : μ ({a | ¬f a = b} ∩ sᶜ) = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
obs : μ (f ⁻¹' t ∩ sᶜ) = 0
⊢ μ (f ⁻¹' t ∩ s) + μ (f ⁻¹' t ∩ sᶜ) ≤ μ (f ⁻¹' t ∩ s)
state_after: no goals

tactic: apply le_antisymm _ (zero_le _)
state_before: R : Type u_1
α : Type u_2
β✝ : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β✝
inst✝¹ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t✝ : Set α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : μ ({a | ¬f a = b} ∩ sᶜ) = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
⊢ μ (f ⁻¹' t ∩ sᶜ) = 0
state_after: R : Type u_1
α : Type u_2
β✝ : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β✝
inst✝¹ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t✝ : Set α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : μ ({a | ¬f a = b} ∩ sᶜ) = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
⊢ μ (f ⁻¹' t ∩ sᶜ) ≤ 0

tactic: rw [← hs]
state_before: R : Type u_1
α : Type u_2
β✝ : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β✝
inst✝¹ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t✝ : Set α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : μ ({a | ¬f a = b} ∩ sᶜ) = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
⊢ μ (f ⁻¹' t ∩ sᶜ) ≤ 0
state_after: R : Type u_1
α : Type u_2
β✝ : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β✝
inst✝¹ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t✝ : Set α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : μ ({a | ¬f a = b} ∩ sᶜ) = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
⊢ μ (f ⁻¹' t ∩ sᶜ) ≤ μ ({a | ¬f a = b} ∩ sᶜ)

tactic: apply measure_mono (inter_subset_inter_left _ _)
state_before: R : Type u_1
α : Type u_2
β✝ : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β✝
inst✝¹ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t✝ : Set α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : μ ({a | ¬f a = b} ∩ sᶜ) = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
⊢ μ (f ⁻¹' t ∩ sᶜ) ≤ μ ({a | ¬f a = b} ∩ sᶜ)
state_after: R : Type u_1
α : Type u_2
β✝ : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β✝
inst✝¹ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t✝ : Set α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : μ ({a | ¬f a = b} ∩ sᶜ) = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
⊢ f ⁻¹' t ⊆ {a | ¬f a = b}

tactic: intro x hx hfx
state_before: R : Type u_1
α : Type u_2
β✝ : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β✝
inst✝¹ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t✝ : Set α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : μ ({a | ¬f a = b} ∩ sᶜ) = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
⊢ f ⁻¹' t ⊆ {a | ¬f a = b}
state_after: R : Type u_1
α : Type u_2
β✝ : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β✝
inst✝¹ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t✝ : Set α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : μ ({a | ¬f a = b} ∩ sᶜ) = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
x : α
hx : x ∈ f ⁻¹' t
hfx : f x = b
⊢ False

tactic: simp only [mem_preimage, mem_setOf_eq] at hx hfx
state_before: R : Type u_1
α : Type u_2
β✝ : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β✝
inst✝¹ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t✝ : Set α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : μ ({a | ¬f a = b} ∩ sᶜ) = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
x : α
hx : x ∈ f ⁻¹' t
hfx : f x = b
⊢ False
state_after: R : Type u_1
α : Type u_2
β✝ : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β✝
inst✝¹ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t✝ : Set α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : μ ({a | ¬f a = b} ∩ sᶜ) = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
x : α
hfx : f x = b
hx : f x ∈ t
⊢ False

tactic: exact ht (hfx ▸ hx)
state_before: R : Type u_1
α : Type u_2
β✝ : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β✝
inst✝¹ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t✝ : Set α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : μ ({a | ¬f a = b} ∩ sᶜ) = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
x : α
hfx : f x = b
hx : f x ∈ t
⊢ False
state_after: no goals

tactic: exact NullMeasurableSet.of_null hs
state_before: R : Type u_1
α : Type u_2
β✝ : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β✝
inst✝¹ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t✝ : Set α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : (μ.restrict sᶜ) {a | ¬f a = b} = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
⊢ NullMeasurableSet {a | ¬f a = b} (μ.restrict sᶜ)
state_after: no goals