l3lab/ntp-mathlib huggingface dataset:
/-- The Lebesgue integral of `g` with respect to the measure `μ.withDensity f` coincides with
the integral of `f * g`. This version assumes that `g` is almost everywhere measurable. For a
version without conditions on `g` but requiring that `f` is almost everywhere finite, see
`lintegral_withDensity_eq_lintegral_mul_non_measurable` -/
theorem lintegral_withDensity_eq_lintegral_mul₀' {μ : Measure α} {f : α → ℝ≥0∞}
    (hf : AEMeasurable f μ) {g : α → ℝ≥0∞} (hg : AEMeasurable g (μ.withDensity f)) :
    ∫⁻ a, g a ∂μ.withDensity f = ∫⁻ a, (f * g) a ∂μ := by
  let f' := hf.mk f
  have : μ.withDensity f = μ.withDensity f' := withDensity_congr_ae hf.ae_eq_mk
  rw [this] at hg ⊢
  let g' := hg.mk g
  calc
    ∫⁻ a, g a ∂μ.withDensity f' = ∫⁻ a, g' a ∂μ.withDensity f' := lintegral_congr_ae hg.ae_eq_mk
    _ = ∫⁻ a, (f' * g') a ∂μ :=
      (lintegral_withDensity_eq_lintegral_mul _ hf.measurable_mk hg.measurable_mk)
    _ = ∫⁻ a, (f' * g) a ∂μ := by
      apply lintegral_congr_ae
      apply ae_of_ae_restrict_of_ae_restrict_compl { x | f' x ≠ 0 }
      · have Z := hg.ae_eq_mk
        rw [EventuallyEq, ae_withDensity_iff_ae_restrict hf.measurable_mk] at Z
        filter_upwards [Z]
        intro x hx
        simp only [hx, Pi.mul_apply]
      · have M : MeasurableSet { x : α | f' x ≠ 0 }ᶜ :=
          (hf.measurable_mk (measurableSet_singleton 0).compl).compl
        

Proof state:
case h.htc
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f
g : α → ℝ≥0∞
f' : α → ℝ≥0∞ := AEMeasurable.mk f hf
hg : AEMeasurable g
this : withDensity μ f = withDensity μ f'
g' : α → ℝ≥0∞ := AEMeasurable.mk g hg
M : MeasurableSet {x | f' x ≠ 0}ᶜ
⊢ ∀ᵐ (x : α) ∂Measure.restrict μ {x | f' x ≠ 0}ᶜ, (fun a => (f' * g') a) x = (fun a => (f' * g) a) x

Next tactic:
filter_upwards [ae_restrict_mem M]
/-- The Lebesgue integral of `g` with respect to the measure `μ.withDensity f` coincides with
the integral of `f * g`. This version assumes that `g` is almost everywhere measurable. For a
version without conditions on `g` but requiring that `f` is almost everywhere finite, see
`lintegral_withDensity_eq_lintegral_mul_non_measurable` -/
theorem lintegral_withDensity_eq_lintegral_mul₀' {μ : Measure α} {f : α → ℝ≥0∞}
    (hf : AEMeasurable f μ) {g : α → ℝ≥0∞} (hg : AEMeasurable g (μ.withDensity f)) :
    ∫⁻ a, g a ∂μ.withDensity f = ∫⁻ a, (f * g) a ∂μ := by
  let f' := hf.mk f
  have : μ.withDensity f = μ.withDensity f' := withDensity_congr_ae hf.ae_eq_mk
  rw [this] at hg ⊢
  let g' := hg.mk g
  calc
    ∫⁻ a, g a ∂μ.withDensity f' = ∫⁻ a, g' a ∂μ.withDensity f' := lintegral_congr_ae hg.ae_eq_mk
    _ = ∫⁻ a, (f' * g') a ∂μ :=
      (lintegral_withDensity_eq_lintegral_mul _ hf.measurable_mk hg.measurable_mk)
    _ = ∫⁻ a, (f' * g) a ∂μ := by
      apply lintegral_congr_ae
      apply ae_of_ae_restrict_of_ae_restrict_compl { x | f' x ≠ 0 }
      · have Z := hg.ae_eq_mk
        rw [EventuallyEq, ae_withDensity_iff_ae_restrict hf.measurable_mk] at Z
        filter_upwards [Z]
        intro x hx
        simp only [hx, Pi.mul_apply]
      · have M : MeasurableSet { x : α | f' x ≠ 0 }ᶜ :=
          (hf.measurable_mk (measurableSet_singleton 0).compl).compl
        filter_upwards [ae_restrict_mem M]
        

Proof state:
case h
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f
g : α → ℝ≥0∞
f' : α → ℝ≥0∞ := AEMeasurable.mk f hf
hg : AEMeasurable g
this : withDensity μ f = withDensity μ f'
g' : α → ℝ≥0∞ := AEMeasurable.mk g hg
M : MeasurableSet {x | f' x ≠ 0}ᶜ
⊢ ∀ a ∈ {x | f' x ≠ 0}ᶜ, (f' * g') a = (f' * g) a

Next tactic:
intro x hx
/-- The Lebesgue integral of `g` with respect to the measure `μ.withDensity f` coincides with
the integral of `f * g`. This version assumes that `g` is almost everywhere measurable. For a
version without conditions on `g` but requiring that `f` is almost everywhere finite, see
`lintegral_withDensity_eq_lintegral_mul_non_measurable` -/
theorem lintegral_withDensity_eq_lintegral_mul₀' {μ : Measure α} {f : α → ℝ≥0∞}
    (hf : AEMeasurable f μ) {g : α → ℝ≥0∞} (hg : AEMeasurable g (μ.withDensity f)) :
    ∫⁻ a, g a ∂μ.withDensity f = ∫⁻ a, (f * g) a ∂μ := by
  let f' := hf.mk f
  have : μ.withDensity f = μ.withDensity f' := withDensity_congr_ae hf.ae_eq_mk
  rw [this] at hg ⊢
  let g' := hg.mk g
  calc
    ∫⁻ a, g a ∂μ.withDensity f' = ∫⁻ a, g' a ∂μ.withDensity f' := lintegral_congr_ae hg.ae_eq_mk
    _ = ∫⁻ a, (f' * g') a ∂μ :=
      (lintegral_withDensity_eq_lintegral_mul _ hf.measurable_mk hg.measurable_mk)
    _ = ∫⁻ a, (f' * g) a ∂μ := by
      apply lintegral_congr_ae
      apply ae_of_ae_restrict_of_ae_restrict_compl { x | f' x ≠ 0 }
      · have Z := hg.ae_eq_mk
        rw [EventuallyEq, ae_withDensity_iff_ae_restrict hf.measurable_mk] at Z
        filter_upwards [Z]
        intro x hx
        simp only [hx, Pi.mul_apply]
      · have M : MeasurableSet { x : α | f' x ≠ 0 }ᶜ :=
          (hf.measurable_mk (measurableSet_singleton 0).compl).compl
        filter_upwards [ae_restrict_mem M]
        intro x hx
        

Proof state:
case h
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f
g : α → ℝ≥0∞
f' : α → ℝ≥0∞ := AEMeasurable.mk f hf
hg : AEMeasurable g
this : withDensity μ f = withDensity μ f'
g' : α → ℝ≥0∞ := AEMeasurable.mk g hg
M : MeasurableSet {x | f' x ≠ 0}ᶜ
x : α
hx : x ∈ {x | f' x ≠ 0}ᶜ
⊢ (f' * g') x = (f' * g) x

Next tactic:
simp only [Classical.not_not, mem_setOf_eq, mem_compl_iff] at hx
/-- The Lebesgue integral of `g` with respect to the measure `μ.withDensity f` coincides with
the integral of `f * g`. This version assumes that `g` is almost everywhere measurable. For a
version without conditions on `g` but requiring that `f` is almost everywhere finite, see
`lintegral_withDensity_eq_lintegral_mul_non_measurable` -/
theorem lintegral_withDensity_eq_lintegral_mul₀' {μ : Measure α} {f : α → ℝ≥0∞}
    (hf : AEMeasurable f μ) {g : α → ℝ≥0∞} (hg : AEMeasurable g (μ.withDensity f)) :
    ∫⁻ a, g a ∂μ.withDensity f = ∫⁻ a, (f * g) a ∂μ := by
  let f' := hf.mk f
  have : μ.withDensity f = μ.withDensity f' := withDensity_congr_ae hf.ae_eq_mk
  rw [this] at hg ⊢
  let g' := hg.mk g
  calc
    ∫⁻ a, g a ∂μ.withDensity f' = ∫⁻ a, g' a ∂μ.withDensity f' := lintegral_congr_ae hg.ae_eq_mk
    _ = ∫⁻ a, (f' * g') a ∂μ :=
      (lintegral_withDensity_eq_lintegral_mul _ hf.measurable_mk hg.measurable_mk)
    _ = ∫⁻ a, (f' * g) a ∂μ := by
      apply lintegral_congr_ae
      apply ae_of_ae_restrict_of_ae_restrict_compl { x | f' x ≠ 0 }
      · have Z := hg.ae_eq_mk
        rw [EventuallyEq, ae_withDensity_iff_ae_restrict hf.measurable_mk] at Z
        filter_upwards [Z]
        intro x hx
        simp only [hx, Pi.mul_apply]
      · have M : MeasurableSet { x : α | f' x ≠ 0 }ᶜ :=
          (hf.measurable_mk (measurableSet_singleton 0).compl).compl
        filter_upwards [ae_restrict_mem M]
        intro x hx
        simp only [Classical.not_not, mem_setOf_eq, mem_compl_iff] at hx
        

Proof state:
case h
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f
g : α → ℝ≥0∞
f' : α → ℝ≥0∞ := AEMeasurable.mk f hf
hg : AEMeasurable g
this : withDensity μ f = withDensity μ f'
g' : α → ℝ≥0∞ := AEMeasurable.mk g hg
M : MeasurableSet {x | f' x ≠ 0}ᶜ
x : α
hx : AEMeasurable.mk f hf x = 0
⊢ (f' * g') x = (f' * g) x

Next tactic:
simp only [hx, zero_mul, Pi.mul_apply]
/-- The Lebesgue integral of `g` with respect to the measure `μ.withDensity f` coincides with
the integral of `f * g`. This version assumes that `g` is almost everywhere measurable. For a
version without conditions on `g` but requiring that `f` is almost everywhere finite, see
`lintegral_withDensity_eq_lintegral_mul_non_measurable` -/
theorem lintegral_withDensity_eq_lintegral_mul₀' {μ : Measure α} {f : α → ℝ≥0∞}
    (hf : AEMeasurable f μ) {g : α → ℝ≥0∞} (hg : AEMeasurable g (μ.withDensity f)) :
    ∫⁻ a, g a ∂μ.withDensity f = ∫⁻ a, (f * g) a ∂μ := by
  let f' := hf.mk f
  have : μ.withDensity f = μ.withDensity f' := withDensity_congr_ae hf.ae_eq_mk
  rw [this] at hg ⊢
  let g' := hg.mk g
  calc
    ∫⁻ a, g a ∂μ.withDensity f' = ∫⁻ a, g' a ∂μ.withDensity f' := lintegral_congr_ae hg.ae_eq_mk
    _ = ∫⁻ a, (f' * g') a ∂μ :=
      (lintegral_withDensity_eq_lintegral_mul _ hf.measurable_mk hg.measurable_mk)
    _ = ∫⁻ a, (f' * g) a ∂μ := by
      apply lintegral_congr_ae
      apply ae_of_ae_restrict_of_ae_restrict_compl { x | f' x ≠ 0 }
      · have Z := hg.ae_eq_mk
        rw [EventuallyEq, ae_withDensity_iff_ae_restrict hf.measurable_mk] at Z
        filter_upwards [Z]
        intro x hx
        simp only [hx, Pi.mul_apply]
      · have M : MeasurableSet { x : α | f' x ≠ 0 }ᶜ :=
          (hf.measurable_mk (measurableSet_singleton 0).compl).compl
        filter_upwards [ae_restrict_mem M]
        intro x hx
        simp only [Classical.not_not, mem_setOf_eq, mem_compl_iff] at hx
        simp only [hx, zero_mul, Pi.mul_apply]
    _ = ∫⁻ a : α, (f * g) a ∂μ := by
      

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f
g : α → ℝ≥0∞
f' : α → ℝ≥0∞ := AEMeasurable.mk f hf
hg : AEMeasurable g
this : withDensity μ f = withDensity μ f'
g' : α → ℝ≥0∞ := AEMeasurable.mk g hg
⊢ ∫⁻ (a : α), (f' * g) a ∂μ = ∫⁻ (a : α), (f * g) a ∂μ

Next tactic:
apply lintegral_congr_ae
/-- The Lebesgue integral of `g` with respect to the measure `μ.withDensity f` coincides with
the integral of `f * g`. This version assumes that `g` is almost everywhere measurable. For a
version without conditions on `g` but requiring that `f` is almost everywhere finite, see
`lintegral_withDensity_eq_lintegral_mul_non_measurable` -/
theorem lintegral_withDensity_eq_lintegral_mul₀' {μ : Measure α} {f : α → ℝ≥0∞}
    (hf : AEMeasurable f μ) {g : α → ℝ≥0∞} (hg : AEMeasurable g (μ.withDensity f)) :
    ∫⁻ a, g a ∂μ.withDensity f = ∫⁻ a, (f * g) a ∂μ := by
  let f' := hf.mk f
  have : μ.withDensity f = μ.withDensity f' := withDensity_congr_ae hf.ae_eq_mk
  rw [this] at hg ⊢
  let g' := hg.mk g
  calc
    ∫⁻ a, g a ∂μ.withDensity f' = ∫⁻ a, g' a ∂μ.withDensity f' := lintegral_congr_ae hg.ae_eq_mk
    _ = ∫⁻ a, (f' * g') a ∂μ :=
      (lintegral_withDensity_eq_lintegral_mul _ hf.measurable_mk hg.measurable_mk)
    _ = ∫⁻ a, (f' * g) a ∂μ := by
      apply lintegral_congr_ae
      apply ae_of_ae_restrict_of_ae_restrict_compl { x | f' x ≠ 0 }
      · have Z := hg.ae_eq_mk
        rw [EventuallyEq, ae_withDensity_iff_ae_restrict hf.measurable_mk] at Z
        filter_upwards [Z]
        intro x hx
        simp only [hx, Pi.mul_apply]
      · have M : MeasurableSet { x : α | f' x ≠ 0 }ᶜ :=
          (hf.measurable_mk (measurableSet_singleton 0).compl).compl
        filter_upwards [ae_restrict_mem M]
        intro x hx
        simp only [Classical.not_not, mem_setOf_eq, mem_compl_iff] at hx
        simp only [hx, zero_mul, Pi.mul_apply]
    _ = ∫⁻ a : α, (f * g) a ∂μ := by
      apply lintegral_congr_ae
      

Proof state:
case h
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f
g : α → ℝ≥0∞
f' : α → ℝ≥0∞ := AEMeasurable.mk f hf
hg : AEMeasurable g
this : withDensity μ f = withDensity μ f'
g' : α → ℝ≥0∞ := AEMeasurable.mk g hg
⊢ (fun a => (f' * g) a) =ᶠ[ae μ] fun a => (f * g) a

Next tactic:
filter_upwards [hf.ae_eq_mk]
/-- The Lebesgue integral of `g` with respect to the measure `μ.withDensity f` coincides with
the integral of `f * g`. This version assumes that `g` is almost everywhere measurable. For a
version without conditions on `g` but requiring that `f` is almost everywhere finite, see
`lintegral_withDensity_eq_lintegral_mul_non_measurable` -/
theorem lintegral_withDensity_eq_lintegral_mul₀' {μ : Measure α} {f : α → ℝ≥0∞}
    (hf : AEMeasurable f μ) {g : α → ℝ≥0∞} (hg : AEMeasurable g (μ.withDensity f)) :
    ∫⁻ a, g a ∂μ.withDensity f = ∫⁻ a, (f * g) a ∂μ := by
  let f' := hf.mk f
  have : μ.withDensity f = μ.withDensity f' := withDensity_congr_ae hf.ae_eq_mk
  rw [this] at hg ⊢
  let g' := hg.mk g
  calc
    ∫⁻ a, g a ∂μ.withDensity f' = ∫⁻ a, g' a ∂μ.withDensity f' := lintegral_congr_ae hg.ae_eq_mk
    _ = ∫⁻ a, (f' * g') a ∂μ :=
      (lintegral_withDensity_eq_lintegral_mul _ hf.measurable_mk hg.measurable_mk)
    _ = ∫⁻ a, (f' * g) a ∂μ := by
      apply lintegral_congr_ae
      apply ae_of_ae_restrict_of_ae_restrict_compl { x | f' x ≠ 0 }
      · have Z := hg.ae_eq_mk
        rw [EventuallyEq, ae_withDensity_iff_ae_restrict hf.measurable_mk] at Z
        filter_upwards [Z]
        intro x hx
        simp only [hx, Pi.mul_apply]
      · have M : MeasurableSet { x : α | f' x ≠ 0 }ᶜ :=
          (hf.measurable_mk (measurableSet_singleton 0).compl).compl
        filter_upwards [ae_restrict_mem M]
        intro x hx
        simp only [Classical.not_not, mem_setOf_eq, mem_compl_iff] at hx
        simp only [hx, zero_mul, Pi.mul_apply]
    _ = ∫⁻ a : α, (f * g) a ∂μ := by
      apply lintegral_congr_ae
      filter_upwards [hf.ae_eq_mk]
      

Proof state:
case h
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f
g : α → ℝ≥0∞
f' : α → ℝ≥0∞ := AEMeasurable.mk f hf
hg : AEMeasurable g
this : withDensity μ f = withDensity μ f'
g' : α → ℝ≥0∞ := AEMeasurable.mk g hg
⊢ ∀ (a : α), f a = AEMeasurable.mk f hf a → (f' * g) a = (f * g) a

Next tactic:
intro x hx
/-- The Lebesgue integral of `g` with respect to the measure `μ.withDensity f` coincides with
the integral of `f * g`. This version assumes that `g` is almost everywhere measurable. For a
version without conditions on `g` but requiring that `f` is almost everywhere finite, see
`lintegral_withDensity_eq_lintegral_mul_non_measurable` -/
theorem lintegral_withDensity_eq_lintegral_mul₀' {μ : Measure α} {f : α → ℝ≥0∞}
    (hf : AEMeasurable f μ) {g : α → ℝ≥0∞} (hg : AEMeasurable g (μ.withDensity f)) :
    ∫⁻ a, g a ∂μ.withDensity f = ∫⁻ a, (f * g) a ∂μ := by
  let f' := hf.mk f
  have : μ.withDensity f = μ.withDensity f' := withDensity_congr_ae hf.ae_eq_mk
  rw [this] at hg ⊢
  let g' := hg.mk g
  calc
    ∫⁻ a, g a ∂μ.withDensity f' = ∫⁻ a, g' a ∂μ.withDensity f' := lintegral_congr_ae hg.ae_eq_mk
    _ = ∫⁻ a, (f' * g') a ∂μ :=
      (lintegral_withDensity_eq_lintegral_mul _ hf.measurable_mk hg.measurable_mk)
    _ = ∫⁻ a, (f' * g) a ∂μ := by
      apply lintegral_congr_ae
      apply ae_of_ae_restrict_of_ae_restrict_compl { x | f' x ≠ 0 }
      · have Z := hg.ae_eq_mk
        rw [EventuallyEq, ae_withDensity_iff_ae_restrict hf.measurable_mk] at Z
        filter_upwards [Z]
        intro x hx
        simp only [hx, Pi.mul_apply]
      · have M : MeasurableSet { x : α | f' x ≠ 0 }ᶜ :=
          (hf.measurable_mk (measurableSet_singleton 0).compl).compl
        filter_upwards [ae_restrict_mem M]
        intro x hx
        simp only [Classical.not_not, mem_setOf_eq, mem_compl_iff] at hx
        simp only [hx, zero_mul, Pi.mul_apply]
    _ = ∫⁻ a : α, (f * g) a ∂μ := by
      apply lintegral_congr_ae
      filter_upwards [hf.ae_eq_mk]
      intro x hx
      

Proof state:
case h
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f
g : α → ℝ≥0∞
f' : α → ℝ≥0∞ := AEMeasurable.mk f hf
hg : AEMeasurable g
this : withDensity μ f = withDensity μ f'
g' : α → ℝ≥0∞ := AEMeasurable.mk g hg
x : α
hx : f x = AEMeasurable.mk f hf x
⊢ (f' * g) x = (f * g) x

Next tactic:
simp only [hx, Pi.mul_apply]
theorem lintegral_withDensity_le_lintegral_mul (μ : Measure α) {f : α → ℝ≥0∞}
    (f_meas : Measurable f) (g : α → ℝ≥0∞) : (∫⁻ a, g a ∂μ.withDensity f) ≤ ∫⁻ a, (f * g) a ∂μ := by
  

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
g : α → ℝ≥0∞
⊢ ∫⁻ (a : α), g a ∂withDensity μ f ≤ ∫⁻ (a : α), (f * g) a ∂μ

Next tactic:
rw [← iSup_lintegral_measurable_le_eq_lintegral, ← iSup_lintegral_measurable_le_eq_lintegral]
theorem lintegral_withDensity_le_lintegral_mul (μ : Measure α) {f : α → ℝ≥0∞}
    (f_meas : Measurable f) (g : α → ℝ≥0∞) : (∫⁻ a, g a ∂μ.withDensity f) ≤ ∫⁻ a, (f * g) a ∂μ := by
  rw [← iSup_lintegral_measurable_le_eq_lintegral, ← iSup_lintegral_measurable_le_eq_lintegral]
  

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
g : α → ℝ≥0∞
⊢ ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ fun a => g a), ∫⁻ (a : α), g_1 a ∂withDensity μ f ≤
    ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ fun a => (f * g) a), ∫⁻ (a : α), g_1 a ∂μ

Next tactic:
refine' iSup₂_le fun i i_meas => iSup_le fun hi => _
theorem lintegral_withDensity_le_lintegral_mul (μ : Measure α) {f : α → ℝ≥0∞}
    (f_meas : Measurable f) (g : α → ℝ≥0∞) : (∫⁻ a, g a ∂μ.withDensity f) ≤ ∫⁻ a, (f * g) a ∂μ := by
  rw [← iSup_lintegral_measurable_le_eq_lintegral, ← iSup_lintegral_measurable_le_eq_lintegral]
  refine' iSup₂_le fun i i_meas => iSup_le fun hi => _
  

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
g i : α → ℝ≥0∞
i_meas : Measurable i
hi : i ≤ fun a => g a
⊢ ∫⁻ (a : α), i a ∂withDensity μ f ≤
    ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ fun a => (f * g) a), ∫⁻ (a : α), g_1 a ∂μ

Next tactic:
have A : f * i ≤ f * g := fun x => mul_le_mul_left' (hi x) _
theorem lintegral_withDensity_le_lintegral_mul (μ : Measure α) {f : α → ℝ≥0∞}
    (f_meas : Measurable f) (g : α → ℝ≥0∞) : (∫⁻ a, g a ∂μ.withDensity f) ≤ ∫⁻ a, (f * g) a ∂μ := by
  rw [← iSup_lintegral_measurable_le_eq_lintegral, ← iSup_lintegral_measurable_le_eq_lintegral]
  refine' iSup₂_le fun i i_meas => iSup_le fun hi => _
  have A : f * i ≤ f * g := fun x => mul_le_mul_left' (hi x) _
  

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
g i : α → ℝ≥0∞
i_meas : Measurable i
hi : i ≤ fun a => g a
A : f * i ≤ f * g
⊢ ∫⁻ (a : α), i a ∂withDensity μ f ≤
    ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ fun a => (f * g) a), ∫⁻ (a : α), g_1 a ∂μ

Next tactic:
refine' le_iSup₂_of_le (f * i) (f_meas.mul i_meas) _
theorem lintegral_withDensity_le_lintegral_mul (μ : Measure α) {f : α → ℝ≥0∞}
    (f_meas : Measurable f) (g : α → ℝ≥0∞) : (∫⁻ a, g a ∂μ.withDensity f) ≤ ∫⁻ a, (f * g) a ∂μ := by
  rw [← iSup_lintegral_measurable_le_eq_lintegral, ← iSup_lintegral_measurable_le_eq_lintegral]
  refine' iSup₂_le fun i i_meas => iSup_le fun hi => _
  have A : f * i ≤ f * g := fun x => mul_le_mul_left' (hi x) _
  refine' le_iSup₂_of_le (f * i) (f_meas.mul i_meas) _
  

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
g i : α → ℝ≥0∞
i_meas : Measurable i
hi : i ≤ fun a => g a
A : f * i ≤ f * g
⊢ ∫⁻ (a : α), i a ∂withDensity μ f ≤ ⨆ (_ : f * i ≤ fun a => (f * g) a), ∫⁻ (a : α), (f * i) a ∂μ

Next tactic:
exact le_iSup_of_le A (le_of_eq (lintegral_withDensity_eq_lintegral_mul _ f_meas i_meas))
theorem lintegral_withDensity_eq_lintegral_mul_non_measurable (μ : Measure α) {f : α → ℝ≥0∞}
    (f_meas : Measurable f) (hf : ∀ᵐ x ∂μ, f x < ∞) (g : α → ℝ≥0∞) :
    ∫⁻ a, g a ∂μ.withDensity f = ∫⁻ a, (f * g) a ∂μ := by
  

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
hf : ∀ᵐ (x : α) ∂μ, f x < ⊤
g : α → ℝ≥0∞
⊢ ∫⁻ (a : α), g a ∂withDensity μ f = ∫⁻ (a : α), (f * g) a ∂μ

Next tactic:
refine' le_antisymm (lintegral_withDensity_le_lintegral_mul μ f_meas g) _
theorem lintegral_withDensity_eq_lintegral_mul_non_measurable (μ : Measure α) {f : α → ℝ≥0∞}
    (f_meas : Measurable f) (hf : ∀ᵐ x ∂μ, f x < ∞) (g : α → ℝ≥0∞) :
    ∫⁻ a, g a ∂μ.withDensity f = ∫⁻ a, (f * g) a ∂μ := by
  refine' le_antisymm (lintegral_withDensity_le_lintegral_mul μ f_meas g) _
  

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
hf : ∀ᵐ (x : α) ∂μ, f x < ⊤
g : α → ℝ≥0∞
⊢ ∫⁻ (a : α), (f * g) a ∂μ ≤ ∫⁻ (a : α), g a ∂withDensity μ f

Next tactic:
rw [← iSup_lintegral_measurable_le_eq_lintegral, ← iSup_lintegral_measurable_le_eq_lintegral]
theorem lintegral_withDensity_eq_lintegral_mul_non_measurable (μ : Measure α) {f : α → ℝ≥0∞}
    (f_meas : Measurable f) (hf : ∀ᵐ x ∂μ, f x < ∞) (g : α → ℝ≥0∞) :
    ∫⁻ a, g a ∂μ.withDensity f = ∫⁻ a, (f * g) a ∂μ := by
  refine' le_antisymm (lintegral_withDensity_le_lintegral_mul μ f_meas g) _
  rw [← iSup_lintegral_measurable_le_eq_lintegral, ← iSup_lintegral_measurable_le_eq_lintegral]
  

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
hf : ∀ᵐ (x : α) ∂μ, f x < ⊤
g : α → ℝ≥0∞
⊢ ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ fun a => (f * g) a), ∫⁻ (a : α), g_1 a ∂μ ≤
    ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ fun a => g a), ∫⁻ (a : α), g_1 a ∂withDensity μ f

Next tactic:
refine' iSup₂_le fun i i_meas => iSup_le fun hi => _
theorem lintegral_withDensity_eq_lintegral_mul_non_measurable (μ : Measure α) {f : α → ℝ≥0∞}
    (f_meas : Measurable f) (hf : ∀ᵐ x ∂μ, f x < ∞) (g : α → ℝ≥0∞) :
    ∫⁻ a, g a ∂μ.withDensity f = ∫⁻ a, (f * g) a ∂μ := by
  refine' le_antisymm (lintegral_withDensity_le_lintegral_mul μ f_meas g) _
  rw [← iSup_lintegral_measurable_le_eq_lintegral, ← iSup_lintegral_measurable_le_eq_lintegral]
  refine' iSup₂_le fun i i_meas => iSup_le fun hi => _
  

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
hf : ∀ᵐ (x : α) ∂μ, f x < ⊤
g i : α → ℝ≥0∞
i_meas : Measurable i
hi : i ≤ fun a => (f * g) a
⊢ ∫⁻ (a : α), i a ∂μ ≤ ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ fun a => g a), ∫⁻ (a : α), g_1 a ∂withDensity μ f

Next tactic:
have A : (fun x => (f x)⁻¹ * i x) ≤ g := by
    intro x
    dsimp
    rw [mul_comm, ← div_eq_mul_inv]
    exact div_le_of_le_mul' (hi x)
theorem lintegral_withDensity_eq_lintegral_mul_non_measurable (μ : Measure α) {f : α → ℝ≥0∞}
    (f_meas : Measurable f) (hf : ∀ᵐ x ∂μ, f x < ∞) (g : α → ℝ≥0∞) :
    ∫⁻ a, g a ∂μ.withDensity f = ∫⁻ a, (f * g) a ∂μ := by
  refine' le_antisymm (lintegral_withDensity_le_lintegral_mul μ f_meas g) _
  rw [← iSup_lintegral_measurable_le_eq_lintegral, ← iSup_lintegral_measurable_le_eq_lintegral]
  refine' iSup₂_le fun i i_meas => iSup_le fun hi => _
  have A : (fun x => (f x)⁻¹ * i x) ≤ g := by
    

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
hf : ∀ᵐ (x : α) ∂μ, f x < ⊤
g i : α → ℝ≥0∞
i_meas : Measurable i
hi : i ≤ fun a => (f * g) a
⊢ (fun x => (f x)⁻¹ * i x) ≤ g

Next tactic:
intro x
theorem lintegral_withDensity_eq_lintegral_mul_non_measurable (μ : Measure α) {f : α → ℝ≥0∞}
    (f_meas : Measurable f) (hf : ∀ᵐ x ∂μ, f x < ∞) (g : α → ℝ≥0∞) :
    ∫⁻ a, g a ∂μ.withDensity f = ∫⁻ a, (f * g) a ∂μ := by
  refine' le_antisymm (lintegral_withDensity_le_lintegral_mul μ f_meas g) _
  rw [← iSup_lintegral_measurable_le_eq_lintegral, ← iSup_lintegral_measurable_le_eq_lintegral]
  refine' iSup₂_le fun i i_meas => iSup_le fun hi => _
  have A : (fun x => (f x)⁻¹ * i x) ≤ g := by
    intro x
    

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
hf : ∀ᵐ (x : α) ∂μ, f x < ⊤
g i : α → ℝ≥0∞
i_meas : Measurable i
hi : i ≤ fun a => (f * g) a
x : α
⊢ (fun x => (f x)⁻¹ * i x) x ≤ g x

Next tactic:
dsimp
theorem lintegral_withDensity_eq_lintegral_mul_non_measurable (μ : Measure α) {f : α → ℝ≥0∞}
    (f_meas : Measurable f) (hf : ∀ᵐ x ∂μ, f x < ∞) (g : α → ℝ≥0∞) :
    ∫⁻ a, g a ∂μ.withDensity f = ∫⁻ a, (f * g) a ∂μ := by
  refine' le_antisymm (lintegral_withDensity_le_lintegral_mul μ f_meas g) _
  rw [← iSup_lintegral_measurable_le_eq_lintegral, ← iSup_lintegral_measurable_le_eq_lintegral]
  refine' iSup₂_le fun i i_meas => iSup_le fun hi => _
  have A : (fun x => (f x)⁻¹ * i x) ≤ g := by
    intro x
    dsimp
    

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
hf : ∀ᵐ (x : α) ∂μ, f x < ⊤
g i : α → ℝ≥0∞
i_meas : Measurable i
hi : i ≤ fun a => (f * g) a
x : α
⊢ (f x)⁻¹ * i x ≤ g x

Next tactic:
rw [mul_comm, ← div_eq_mul_inv]