Theorem: MeasureTheory.ae_lt_top
File: Mathlib/MeasureTheory/Integral/Lebesgue.lean

tactic: simp_rw [ae_iff, ENNReal.not_lt_top]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : α → ℝ≥0∞
hf : Measurable f
h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤
⊢ ∀ᵐ (x : α) ∂μ, f x < ⊤
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : α → ℝ≥0∞
hf : Measurable f
h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤
⊢ μ {a | f a = ⊤} = 0

tactic: by_contra h
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : α → ℝ≥0∞
hf : Measurable f
h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤
⊢ μ {a | f a = ⊤} = 0
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : α → ℝ≥0∞
hf : Measurable f
h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤
h : ¬μ {a | f a = ⊤} = 0
⊢ False

tactic: apply h2f.lt_top.not_le
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : α → ℝ≥0∞
hf : Measurable f
h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤
h : ¬μ {a | f a = ⊤} = 0
⊢ False
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : α → ℝ≥0∞
hf : Measurable f
h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤
h : ¬μ {a | f a = ⊤} = 0
⊢ ⊤ ≤ ∫⁻ (x : α), f x ∂μ

tactic: have : (f ⁻¹' {∞}).indicator ⊤ ≤ f := by
  intro x
  by_cases hx : x ∈ f ⁻¹' {∞} <;> [simpa [indicator_of_mem hx]; simp [indicator_of_not_mem hx]]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : α → ℝ≥0∞
hf : Measurable f
h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤
h : ¬μ {a | f a = ⊤} = 0
⊢ ⊤ ≤ ∫⁻ (x : α), f x ∂μ
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : α → ℝ≥0∞
hf : Measurable f
h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤
h : ¬μ {a | f a = ⊤} = 0
this : (f ⁻¹' {⊤}).indicator ⊤ ≤ f
⊢ ⊤ ≤ ∫⁻ (x : α), f x ∂μ

tactic: convert lintegral_mono this
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : α → ℝ≥0∞
hf : Measurable f
h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤
h : ¬μ {a | f a = ⊤} = 0
this : (f ⁻¹' {⊤}).indicator ⊤ ≤ f
⊢ ⊤ ≤ ∫⁻ (x : α), f x ∂μ
state_after: case h.e'_3
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : α → ℝ≥0∞
hf : Measurable f
h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤
h : ¬μ {a | f a = ⊤} = 0
this : (f ⁻¹' {⊤}).indicator ⊤ ≤ f
⊢ ⊤ = ∫⁻ (a : α), (f ⁻¹' {⊤}).indicator ⊤ a ∂μ

tactic: rw [lintegral_indicator _ (hf (measurableSet_singleton ∞))]
state_before: case h.e'_3
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : α → ℝ≥0∞
hf : Measurable f
h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤
h : ¬μ {a | f a = ⊤} = 0
this : (f ⁻¹' {⊤}).indicator ⊤ ≤ f
⊢ ⊤ = ∫⁻ (a : α), (f ⁻¹' {⊤}).indicator ⊤ a ∂μ
state_after: case h.e'_3
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : α → ℝ≥0∞
hf : Measurable f
h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤
h : ¬μ {a | f a = ⊤} = 0
this : (f ⁻¹' {⊤}).indicator ⊤ ≤ f
⊢ ⊤ = ∫⁻ (a : α) in f ⁻¹' {⊤}, ⊤ a ∂μ

tactic: simp [ENNReal.top_mul', preimage, h]
state_before: case h.e'_3
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : α → ℝ≥0∞
hf : Measurable f
h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤
h : ¬μ {a | f a = ⊤} = 0
this : (f ⁻¹' {⊤}).indicator ⊤ ≤ f
⊢ ⊤ = ∫⁻ (a : α) in f ⁻¹' {⊤}, ⊤ a ∂μ
state_after: no goals

tactic: intro x
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : α → ℝ≥0∞
hf : Measurable f
h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤
h : ¬μ {a | f a = ⊤} = 0
⊢ (f ⁻¹' {⊤}).indicator ⊤ ≤ f
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : α → ℝ≥0∞
hf : Measurable f
h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤
h : ¬μ {a | f a = ⊤} = 0
x : α
⊢ (f ⁻¹' {⊤}).indicator ⊤ x ≤ f x

tactic: by_cases hx : x ∈ f ⁻¹' {∞} <;> [simpa [indicator_of_mem hx]; simp [indicator_of_not_mem hx]]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : α → ℝ≥0∞
hf : Measurable f
h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤
h : ¬μ {a | f a = ⊤} = 0
x : α
⊢ (f ⁻¹' {⊤}).indicator ⊤ x ≤ f x
state_after: no goals