Theorem: MeasureTheory.pdf.hasFiniteIntegral_mul
File: Mathlib/Probability/Density.lean

tactic: rw [HasFiniteIntegral]
state_before: Ω : Type u_1
E : Type u_2
inst✝¹ : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
inst✝ : IsFiniteMeasure ℙ
X : Ω → ℝ
f : ℝ → ℝ
g : ℝ → ℝ≥0∞
hg : pdf X ℙ volume =ᶠ[ae volume] g
hgi : ∫⁻ (x : ℝ), ↑‖f x‖₊ * g x ≠ ⊤
⊢ HasFiniteIntegral (fun x => f x * (pdf X ℙ volume x).toReal) volume
state_after: Ω : Type u_1
E : Type u_2
inst✝¹ : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
inst✝ : IsFiniteMeasure ℙ
X : Ω → ℝ
f : ℝ → ℝ
g : ℝ → ℝ≥0∞
hg : pdf X ℙ volume =ᶠ[ae volume] g
hgi : ∫⁻ (x : ℝ), ↑‖f x‖₊ * g x ≠ ⊤
⊢ ∫⁻ (a : ℝ), ↑‖f a * (pdf X ℙ volume a).toReal‖₊ < ⊤

tactic: have : (fun x => ↑‖f x‖₊ * g x) =ᵐ[volume] fun x => ‖f x * (pdf X ℙ volume x).toReal‖₊ := by
  refine ae_eq_trans (Filter.EventuallyEq.mul (ae_eq_refl fun x => (‖f x‖₊ : ℝ≥0∞))
    (ae_eq_trans hg.symm ofReal_toReal_ae_eq.symm)) ?_
  simp_rw [← smul_eq_mul, nnnorm_smul, ENNReal.coe_mul, smul_eq_mul]
  refine Filter.EventuallyEq.mul (ae_eq_refl _) ?_
  simp only [Real.ennnorm_eq_ofReal ENNReal.toReal_nonneg, ae_eq_refl]
state_before: Ω : Type u_1
E : Type u_2
inst✝¹ : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
inst✝ : IsFiniteMeasure ℙ
X : Ω → ℝ
f : ℝ → ℝ
g : ℝ → ℝ≥0∞
hg : pdf X ℙ volume =ᶠ[ae volume] g
hgi : ∫⁻ (x : ℝ), ↑‖f x‖₊ * g x ≠ ⊤
⊢ ∫⁻ (a : ℝ), ↑‖f a * (pdf X ℙ volume a).toReal‖₊ < ⊤
state_after: Ω : Type u_1
E : Type u_2
inst✝¹ : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
inst✝ : IsFiniteMeasure ℙ
X : Ω → ℝ
f : ℝ → ℝ
g : ℝ → ℝ≥0∞
hg : pdf X ℙ volume =ᶠ[ae volume] g
hgi : ∫⁻ (x : ℝ), ↑‖f x‖₊ * g x ≠ ⊤
this : (fun x => ↑‖f x‖₊ * g x) =ᶠ[ae volume] fun x => ↑‖f x * (pdf X ℙ volume x).toReal‖₊
⊢ ∫⁻ (a : ℝ), ↑‖f a * (pdf X ℙ volume a).toReal‖₊ < ⊤

tactic: rwa [lt_top_iff_ne_top, ← lintegral_congr_ae this]
state_before: Ω : Type u_1
E : Type u_2
inst✝¹ : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
inst✝ : IsFiniteMeasure ℙ
X : Ω → ℝ
f : ℝ → ℝ
g : ℝ → ℝ≥0∞
hg : pdf X ℙ volume =ᶠ[ae volume] g
hgi : ∫⁻ (x : ℝ), ↑‖f x‖₊ * g x ≠ ⊤
this : (fun x => ↑‖f x‖₊ * g x) =ᶠ[ae volume] fun x => ↑‖f x * (pdf X ℙ volume x).toReal‖₊
⊢ ∫⁻ (a : ℝ), ↑‖f a * (pdf X ℙ volume a).toReal‖₊ < ⊤
state_after: no goals

tactic: refine ae_eq_trans (Filter.EventuallyEq.mul (ae_eq_refl fun x => (‖f x‖₊ : ℝ≥0∞))
  (ae_eq_trans hg.symm ofReal_toReal_ae_eq.symm)) ?_
state_before: Ω : Type u_1
E : Type u_2
inst✝¹ : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
inst✝ : IsFiniteMeasure ℙ
X : Ω → ℝ
f : ℝ → ℝ
g : ℝ → ℝ≥0∞
hg : pdf X ℙ volume =ᶠ[ae volume] g
hgi : ∫⁻ (x : ℝ), ↑‖f x‖₊ * g x ≠ ⊤
⊢ (fun x => ↑‖f x‖₊ * g x) =ᶠ[ae volume] fun x => ↑‖f x * (pdf X ℙ volume x).toReal‖₊
state_after: Ω : Type u_1
E : Type u_2
inst✝¹ : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
inst✝ : IsFiniteMeasure ℙ
X : Ω → ℝ
f : ℝ → ℝ
g : ℝ → ℝ≥0∞
hg : pdf X ℙ volume =ᶠ[ae volume] g
hgi : ∫⁻ (x : ℝ), ↑‖f x‖₊ * g x ≠ ⊤
⊢ (fun x => ↑‖f x‖₊ * ENNReal.ofReal (pdf X ℙ volume x).toReal) =ᶠ[ae volume] fun x =>
    ↑‖f x * (pdf X ℙ volume x).toReal‖₊

tactic: simp_rw [← smul_eq_mul, nnnorm_smul, ENNReal.coe_mul, smul_eq_mul]
state_before: Ω : Type u_1
E : Type u_2
inst✝¹ : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
inst✝ : IsFiniteMeasure ℙ
X : Ω → ℝ
f : ℝ → ℝ
g : ℝ → ℝ≥0∞
hg : pdf X ℙ volume =ᶠ[ae volume] g
hgi : ∫⁻ (x : ℝ), ↑‖f x‖₊ * g x ≠ ⊤
⊢ (fun x => ↑‖f x‖₊ * ENNReal.ofReal (pdf X ℙ volume x).toReal) =ᶠ[ae volume] fun x =>
    ↑‖f x * (pdf X ℙ volume x).toReal‖₊
state_after: Ω : Type u_1
E : Type u_2
inst✝¹ : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
inst✝ : IsFiniteMeasure ℙ
X : Ω → ℝ
f : ℝ → ℝ
g : ℝ → ℝ≥0∞
hg : pdf X ℙ volume =ᶠ[ae volume] g
hgi : ∫⁻ (x : ℝ), ↑‖f x‖₊ * g x ≠ ⊤
⊢ (fun x => ↑‖f x‖₊ * ENNReal.ofReal (pdf X ℙ volume x).toReal) =ᶠ[ae volume] fun x =>
    ↑‖f x‖₊ * ↑‖(pdf X ℙ volume x).toReal‖₊

tactic: refine Filter.EventuallyEq.mul (ae_eq_refl _) ?_
state_before: Ω : Type u_1
E : Type u_2
inst✝¹ : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
inst✝ : IsFiniteMeasure ℙ
X : Ω → ℝ
f : ℝ → ℝ
g : ℝ → ℝ≥0∞
hg : pdf X ℙ volume =ᶠ[ae volume] g
hgi : ∫⁻ (x : ℝ), ↑‖f x‖₊ * g x ≠ ⊤
⊢ (fun x => ↑‖f x‖₊ * ENNReal.ofReal (pdf X ℙ volume x).toReal) =ᶠ[ae volume] fun x =>
    ↑‖f x‖₊ * ↑‖(pdf X ℙ volume x).toReal‖₊
state_after: Ω : Type u_1
E : Type u_2
inst✝¹ : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
inst✝ : IsFiniteMeasure ℙ
X : Ω → ℝ
f : ℝ → ℝ
g : ℝ → ℝ≥0∞
hg : pdf X ℙ volume =ᶠ[ae volume] g
hgi : ∫⁻ (x : ℝ), ↑‖f x‖₊ * g x ≠ ⊤
⊢ (fun x => ENNReal.ofReal (pdf X ℙ volume x).toReal) =ᶠ[ae volume] fun x => ↑‖(pdf X ℙ volume x).toReal‖₊

tactic: simp only [Real.ennnorm_eq_ofReal ENNReal.toReal_nonneg, ae_eq_refl]
state_before: Ω : Type u_1
E : Type u_2
inst✝¹ : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
inst✝ : IsFiniteMeasure ℙ
X : Ω → ℝ
f : ℝ → ℝ
g : ℝ → ℝ≥0∞
hg : pdf X ℙ volume =ᶠ[ae volume] g
hgi : ∫⁻ (x : ℝ), ↑‖f x‖₊ * g x ≠ ⊤
⊢ (fun x => ENNReal.ofReal (pdf X ℙ volume x).toReal) =ᶠ[ae volume] fun x => ↑‖(pdf X ℙ volume x).toReal‖₊
state_after: no goals