ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.withDensity_apply'
theorem withDensity_apply' [SFinite μ] (f : α → ℝ≥0∞) (s : Set α) :
    μ.withDensity f s = ∫⁻ a in s, f a ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.SFinite μ] (f : α → ℝ≥0∞) (s : Set α)
Goal: (μ.withDensity f) s = ∫⁻ (a : α) in s, f a ∂μ
-- MeasureTheory.withDensity_zero_left
@[simp]
lemma withDensity_zero_left (f : α → ℝ≥0∞) : (0 : Measure α).withDensity f = 0 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (f : α → ℝ≥0∞)
Goal: MeasureTheory.Measure.withDensity 0 f = 0
-- MeasureTheory.withDensity_congr_ae
theorem withDensity_congr_ae {f g : α → ℝ≥0∞} (h : f =ᵐ[μ] g) :
    μ.withDensity f = μ.withDensity g 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ℝ≥0∞} (h : f =ᵐ[μ] g)
Goal: μ.withDensity f = μ.withDensity g
-- MeasureTheory.withDensity_mono
lemma withDensity_mono {f g : α → ℝ≥0∞} (hfg : f ≤ᵐ[μ] g) :
    μ.withDensity f ≤ μ.withDensity g 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ℝ≥0∞} (hfg : f ≤ᵐ[μ] g)
Goal: μ.withDensity f ≤ μ.withDensity g
-- MeasureTheory.withDensity_add_left
theorem withDensity_add_left {f : α → ℝ≥0∞} (hf : Measurable f) (g : α → ℝ≥0∞) :
    μ.withDensity (f + g) = μ.withDensity f + μ.withDensity g 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : Measurable f) (g : α → ℝ≥0∞)
Goal: μ.withDensity (f + g) = μ.withDensity f + μ.withDensity g
-- MeasureTheory.withDensity_add_right
theorem withDensity_add_right (f : α → ℝ≥0∞) {g : α → ℝ≥0∞} (hg : Measurable g) :
    μ.withDensity (f + g) = μ.withDensity f + μ.withDensity g 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ℝ≥0∞) {g : α → ℝ≥0∞} (hg : Measurable g)
Goal: μ.withDensity (f + g) = μ.withDensity f + μ.withDensity g
-- MeasureTheory.withDensity_add_measure
theorem withDensity_add_measure {m : MeasurableSpace α} (μ ν : Measure α) (f : α → ℝ≥0∞) :
    (μ + ν).withDensity f = μ.withDensity f + ν.withDensity f 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ ν : MeasureTheory.Measure α) (f : α → ℝ≥0∞)
Goal: (μ + ν).withDensity f = μ.withDensity f + ν.withDensity f
-- MeasureTheory.withDensity_sum
theorem withDensity_sum {ι : Type*} {m : MeasurableSpace α} (μ : ι → Measure α) (f : α → ℝ≥0∞) :
    (sum μ).withDensity f = sum fun n => (μ n).withDensity f 
Arguments: {α : Type u_1} {ι : Type u_2} {m : MeasurableSpace α} (μ : ι → MeasureTheory.Measure α) (f : α → ℝ≥0∞)
Goal: (MeasureTheory.Measure.sum μ).withDensity f = MeasureTheory.Measure.sum fun n => (μ n).withDensity f
-- MeasureTheory.withDensity_smul
theorem withDensity_smul (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : Measurable f) :
    μ.withDensity (r • f) = r • μ.withDensity f 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : Measurable f)
Goal: μ.withDensity (r • f) = r • μ.withDensity f
-- MeasureTheory.withDensity_smul'
theorem withDensity_smul' (r : ℝ≥0∞) (f : α → ℝ≥0∞) (hr : r ≠ ∞) :
    μ.withDensity (r • f) = r • μ.withDensity f 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (r : ℝ≥0∞) (f : α → ℝ≥0∞) (hr : r ≠ ⊤)
Goal: μ.withDensity (r • f) = r • μ.withDensity f
-- MeasureTheory.withDensity_smul_measure
theorem withDensity_smul_measure (r : ℝ≥0∞) (f : α → ℝ≥0∞) :
    (r • μ).withDensity f = r • μ.withDensity f 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (r : ℝ≥0∞) (f : α → ℝ≥0∞)
Goal: (r • μ).withDensity f = r • μ.withDensity f
-- MeasureTheory.isFiniteMeasure_withDensity
theorem isFiniteMeasure_withDensity {f : α → ℝ≥0∞} (hf : ∫⁻ a, f a ∂μ ≠ ∞) :
    IsFiniteMeasure (μ.withDensity f) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : ∫⁻ (a : α), f a ∂μ ≠ ⊤)
Goal: MeasureTheory.IsFiniteMeasure (μ.withDensity f)
-- MeasureTheory.withDensity_absolutelyContinuous
theorem withDensity_absolutelyContinuous {m : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥0∞) :
    μ.withDensity f ≪ μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (f : α → ℝ≥0∞)
Goal: (μ.withDensity f).AbsolutelyContinuous μ
-- MeasureTheory.withDensity_zero
@[simp]
theorem withDensity_zero : μ.withDensity 0 = 0 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: μ.withDensity 0 = 0
-- MeasureTheory.withDensity_one
@[simp]
theorem withDensity_one : μ.withDensity 1 = μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: μ.withDensity 1 = μ
-- MeasureTheory.withDensity_const
@[simp]
theorem withDensity_const (c : ℝ≥0∞) : μ.withDensity (fun _ ↦ c) = c • μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (c : ℝ≥0∞)
Goal: (μ.withDensity fun x => c) = c • μ
-- MeasureTheory.withDensity_tsum
theorem withDensity_tsum {ι : Type*} [Countable ι] {f : ι → α → ℝ≥0∞} (h : ∀ i, Measurable (f i)) :
    μ.withDensity (∑' n, f n) = sum fun n => μ.withDensity (f n) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Type u_2} [Countable ι] {f : ι → α → ℝ≥0∞} (h : ∀ (i : ι), Measurable (f i))
Goal: μ.withDensity (∑' (n : ι), f n) = MeasureTheory.Measure.sum fun n => μ.withDensity (f n)
-- MeasureTheory.withDensity_indicator
theorem withDensity_indicator {s : Set α} (hs : MeasurableSet s) (f : α → ℝ≥0∞) :
    μ.withDensity (s.indicator f) = (μ.restrict s).withDensity f 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s) (f : α → ℝ≥0∞)
Goal: μ.withDensity (s.indicator f) = (μ.restrict s).withDensity f
-- MeasureTheory.withDensity_indicator_one
theorem withDensity_indicator_one {s : Set α} (hs : MeasurableSet s) :
    μ.withDensity (s.indicator 1) = μ.restrict s 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s)
Goal: μ.withDensity (s.indicator 1) = μ.restrict s
-- MeasureTheory.withDensity_ofReal_mutuallySingular
theorem withDensity_ofReal_mutuallySingular {f : α → ℝ} (hf : Measurable f) :
    (μ.withDensity fun x => ENNReal.ofReal <| f x) ⟂ₘ
      μ.withDensity fun x => ENNReal.ofReal <| -f x 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : Measurable f)
Goal: (μ.withDensity fun x => ENNReal.ofReal (f x)).MutuallySingular (μ.withDensity fun x => ENNReal.ofReal (-f x))
-- MeasureTheory.restrict_withDensity
theorem restrict_withDensity {s : Set α} (hs : MeasurableSet s) (f : α → ℝ≥0∞) :
    (μ.withDensity f).restrict s = (μ.restrict s).withDensity f 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s) (f : α → ℝ≥0∞)
Goal: (μ.withDensity f).restrict s = (μ.restrict s).withDensity f
-- MeasureTheory.restrict_withDensity'
theorem restrict_withDensity' [SFinite μ] (s : Set α) (f : α → ℝ≥0∞) :
    (μ.withDensity f).restrict s = (μ.restrict s).withDensity f 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.SFinite μ] (s : Set α) (f : α → ℝ≥0∞)
Goal: (μ.withDensity f).restrict s = (μ.restrict s).withDensity f
-- MeasureTheory.trim_withDensity
lemma trim_withDensity {m m0 : MeasurableSpace α} {μ : Measure α}
    (hm : m ≤ m0) {f : α → ℝ≥0∞} (hf : Measurable[m] f) :
    (μ.withDensity f).trim hm = (μ.trim hm).withDensity f 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) {f : α → ℝ≥0∞} (hf : Measurable f)
Goal: (μ.withDensity f).trim hm = (μ.trim hm).withDensity f
-- MeasureTheory.Measure.MutuallySingular.withDensity
lemma Measure.MutuallySingular.withDensity {ν : Measure α} {f : α → ℝ≥0∞} (h : μ ⟂ₘ ν) :
    μ.withDensity f ⟂ₘ ν 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (h : μ.MutuallySingular ν)
Goal: (μ.withDensity f).MutuallySingular ν
-- MeasureTheory.withDensity_eq_zero_iff
@[simp]
theorem withDensity_eq_zero_iff {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) :
    μ.withDensity f = 0 ↔ f =ᵐ[μ] 0 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : AEMeasurable f μ)
Goal: μ.withDensity f = 0 ↔ f =ᵐ[μ] 0
-- MeasureTheory.withDensity_apply_eq_zero'
theorem withDensity_apply_eq_zero' {f : α → ℝ≥0∞} {s : Set α} (hf : AEMeasurable f μ) :
    μ.withDensity f s = 0 ↔ μ ({ x | f x ≠ 0 } ∩ s) = 0 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} {s : Set α} (hf : AEMeasurable f μ)
Goal: (μ.withDensity f) s = 0 ↔ μ ({x | f x ≠ 0} ∩ s) = 0
-- MeasureTheory.withDensity_apply_eq_zero
theorem withDensity_apply_eq_zero {f : α → ℝ≥0∞} {s : Set α} (hf : Measurable f) :
    μ.withDensity f s = 0 ↔ μ ({ x | f x ≠ 0 } ∩ s) = 0 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} {s : Set α} (hf : Measurable f)
Goal: (μ.withDensity f) s = 0 ↔ μ ({x | f x ≠ 0} ∩ s) = 0
-- MeasureTheory.ae_withDensity_iff'
theorem ae_withDensity_iff' {p : α → Prop} {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) :
    (∀ᵐ x ∂μ.withDensity f, p x) ↔ ∀ᵐ x ∂μ, f x ≠ 0 → p x 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : α → Prop} {f : α → ℝ≥0∞} (hf : AEMeasurable f μ)
Goal: (∀ᵐ (x : α) ∂μ.withDensity f, p x) ↔ ∀ᵐ (x : α) ∂μ, f x ≠ 0 → p x
-- MeasureTheory.ae_withDensity_iff
theorem ae_withDensity_iff {p : α → Prop} {f : α → ℝ≥0∞} (hf : Measurable f) :
    (∀ᵐ x ∂μ.withDensity f, p x) ↔ ∀ᵐ x ∂μ, f x ≠ 0 → p x 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : α → Prop} {f : α → ℝ≥0∞} (hf : Measurable f)
Goal: (∀ᵐ (x : α) ∂μ.withDensity f, p x) ↔ ∀ᵐ (x : α) ∂μ, f x ≠ 0 → p x
-- MeasureTheory.ae_withDensity_iff_ae_restrict'
theorem ae_withDensity_iff_ae_restrict' {p : α → Prop} {f : α → ℝ≥0∞}
    (hf : AEMeasurable f μ) :
    (∀ᵐ x ∂μ.withDensity f, p x) ↔ ∀ᵐ x ∂μ.restrict { x | f x ≠ 0 }, p x 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : α → Prop} {f : α → ℝ≥0∞} (hf : AEMeasurable f μ)
Goal: (∀ᵐ (x : α) ∂μ.withDensity f, p x) ↔ ∀ᵐ (x : α) ∂μ.restrict {x | f x ≠ 0}, p x
-- MeasureTheory.ae_withDensity_iff_ae_restrict
theorem ae_withDensity_iff_ae_restrict {p : α → Prop} {f : α → ℝ≥0∞} (hf : Measurable f) :
    (∀ᵐ x ∂μ.withDensity f, p x) ↔ ∀ᵐ x ∂μ.restrict { x | f x ≠ 0 }, p x 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : α → Prop} {f : α → ℝ≥0∞} (hf : Measurable f)
Goal: (∀ᵐ (x : α) ∂μ.withDensity f, p x) ↔ ∀ᵐ (x : α) ∂μ.restrict {x | f x ≠ 0}, p x
-- MeasureTheory.aemeasurable_withDensity_ennreal_iff'
theorem aemeasurable_withDensity_ennreal_iff' {f : α → ℝ≥0}
    (hf : AEMeasurable f μ) {g : α → ℝ≥0∞} :
    AEMeasurable g (μ.withDensity fun x => (f x : ℝ≥0∞)) ↔
      AEMeasurable (fun x => (f x : ℝ≥0∞) * g x) μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0} (hf : AEMeasurable f μ) {g : α → ℝ≥0∞}
Goal: AEMeasurable g (μ.withDensity fun x => ↑(f x)) ↔ AEMeasurable (fun x => ↑(f x) * g x) μ
-- MeasureTheory.aemeasurable_withDensity_ennreal_iff
theorem aemeasurable_withDensity_ennreal_iff {f : α → ℝ≥0} (hf : Measurable f) {g : α → ℝ≥0∞} :
    AEMeasurable g (μ.withDensity fun x => (f x : ℝ≥0∞)) ↔
      AEMeasurable (fun x => (f x : ℝ≥0∞) * g x) μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0} (hf : Measurable f) {g : α → ℝ≥0∞}
Goal: AEMeasurable g (μ.withDensity fun x => ↑(f x)) ↔ AEMeasurable (fun x => ↑(f x) * g x) μ
-- MeasureTheory.lintegral_withDensity_eq_lintegral_mul
theorem lintegral_withDensity_eq_lintegral_mul (μ : Measure α) {f : α → ℝ≥0∞}
    (h_mf : Measurable f) :
    ∀ {g : α → ℝ≥0∞}, Measurable g → ∫⁻ a, g a ∂μ.withDensity f = ∫⁻ a, (f * g) a ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) {f : α → ℝ≥0∞} (h_mf : Measurable f) {g : α → ℝ≥0∞}
Goal: Measurable g → ∫⁻ (a : α), g a ∂μ.withDensity f = ∫⁻ (a : α), (f * g) a ∂μ
-- MeasureTheory.setLIntegral_withDensity_eq_setLIntegral_mul
theorem setLIntegral_withDensity_eq_setLIntegral_mul (μ : Measure α) {f g : α → ℝ≥0∞}
    (hf : Measurable f) (hg : Measurable g) {s : Set α} (hs : MeasurableSet s) :
    ∫⁻ x in s, g x ∂μ.withDensity f = ∫⁻ x in s, (f * g) x ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g) {s : Set α} (hs : MeasurableSet s)
Goal: ∫⁻ (x : α) in s, g x ∂μ.withDensity f = ∫⁻ (x : α) in s, (f * g) x ∂μ
-- MeasureTheory.lintegral_withDensity_eq_lintegral_mul₀'
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 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) {g : α → ℝ≥0∞} (hg : AEMeasurable g (μ.withDensity f))
Goal: ∫⁻ (a : α), g a ∂μ.withDensity f = ∫⁻ (a : α), (f * g) a ∂μ
-- MeasureTheory.setLIntegral_withDensity_eq_lintegral_mul₀'
lemma setLIntegral_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))
    {s : Set α} (hs : MeasurableSet s) :
    ∫⁻ a in s, g a ∂μ.withDensity f = ∫⁻ a in s, (f * g) a ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) {g : α → ℝ≥0∞} (hg : AEMeasurable g (μ.withDensity f)) {s : Set α} (hs : MeasurableSet s)
Goal: ∫⁻ (a : α) in s, g a ∂μ.withDensity f = ∫⁻ (a : α) in s, (f * g) a ∂μ
-- MeasureTheory.lintegral_withDensity_eq_lintegral_mul₀
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) :
    ∫⁻ a, g a ∂μ.withDensity f = ∫⁻ a, (f * g) a ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) {g : α → ℝ≥0∞} (hg : AEMeasurable g μ)
Goal: ∫⁻ (a : α), g a ∂μ.withDensity f = ∫⁻ (a : α), (f * g) a ∂μ
-- MeasureTheory.setLIntegral_withDensity_eq_lintegral_mul₀
lemma setLIntegral_withDensity_eq_lintegral_mul₀ {μ : Measure α} {f : α → ℝ≥0∞}
    (hf : AEMeasurable f μ) {g : α → ℝ≥0∞} (hg : AEMeasurable g μ)
    {s : Set α} (hs : MeasurableSet s) :
    ∫⁻ a in s, g a ∂μ.withDensity f = ∫⁻ a in s, (f * g) a ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) {g : α → ℝ≥0∞} (hg : AEMeasurable g μ) {s : Set α} (hs : MeasurableSet s)
Goal: ∫⁻ (a : α) in s, g a ∂μ.withDensity f = ∫⁻ (a : α) in s, (f * g) a ∂μ
-- MeasureTheory.lintegral_withDensity_le_lintegral_mul
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 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) {f : α → ℝ≥0∞} (f_meas : Measurable f) (g : α → ℝ≥0∞)
Goal: ∫⁻ (a : α), g a ∂μ.withDensity f ≤ ∫⁻ (a : α), (f * g) a ∂μ
-- MeasureTheory.lintegral_withDensity_eq_lintegral_mul_non_measurable
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 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) {f : α → ℝ≥0∞} (f_meas : Measurable f) (hf : ∀ᵐ (x : α) ∂μ, f x < ⊤) (g : α → ℝ≥0∞)
Goal: ∫⁻ (a : α), g a ∂μ.withDensity f = ∫⁻ (a : α), (f * g) a ∂μ
-- MeasureTheory.setLIntegral_withDensity_eq_setLIntegral_mul_non_measurable
theorem setLIntegral_withDensity_eq_setLIntegral_mul_non_measurable (μ : Measure α) {f : α → ℝ≥0∞}
    (f_meas : Measurable f) (g : α → ℝ≥0∞) {s : Set α} (hs : MeasurableSet s)
    (hf : ∀ᵐ x ∂μ.restrict s, f x < ∞) :
    ∫⁻ a in s, g a ∂μ.withDensity f = ∫⁻ a in s, (f * g) a ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) {f : α → ℝ≥0∞} (f_meas : Measurable f) (g : α → ℝ≥0∞) {s : Set α} (hs : MeasurableSet s) (hf : ∀ᵐ (x : α) ∂μ.restrict s, f x < ⊤)
Goal: ∫⁻ (a : α) in s, g a ∂μ.withDensity f = ∫⁻ (a : α) in s, (f * g) a ∂μ
-- MeasureTheory.lintegral_withDensity_eq_lintegral_mul_non_measurable₀
theorem lintegral_withDensity_eq_lintegral_mul_non_measurable₀ (μ : Measure α) {f : α → ℝ≥0∞}
    (hf : AEMeasurable f μ) (h'f : ∀ᵐ x ∂μ, f x < ∞) (g : α → ℝ≥0∞) :
    ∫⁻ a, g a ∂μ.withDensity f = ∫⁻ a, (f * g) a ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (h'f : ∀ᵐ (x : α) ∂μ, f x < ⊤) (g : α → ℝ≥0∞)
Goal: ∫⁻ (a : α), g a ∂μ.withDensity f = ∫⁻ (a : α), (f * g) a ∂μ
-- MeasureTheory.setLIntegral_withDensity_eq_setLIntegral_mul_non_measurable₀
theorem setLIntegral_withDensity_eq_setLIntegral_mul_non_measurable₀ (μ : Measure α)
    {f : α → ℝ≥0∞} {s : Set α} (hf : AEMeasurable f (μ.restrict s)) (g : α → ℝ≥0∞)
    (hs : MeasurableSet s) (h'f : ∀ᵐ x ∂μ.restrict s, f x < ∞) :
    ∫⁻ a in s, g a ∂μ.withDensity f = ∫⁻ a in s, (f * g) a ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) {f : α → ℝ≥0∞} {s : Set α} (hf : AEMeasurable f (μ.restrict s)) (g : α → ℝ≥0∞) (hs : MeasurableSet s) (h'f : ∀ᵐ (x : α) ∂μ.restrict s, f x < ⊤)
Goal: ∫⁻ (a : α) in s, g a ∂μ.withDensity f = ∫⁻ (a : α) in s, (f * g) a ∂μ
-- MeasureTheory.setLIntegral_withDensity_eq_setLIntegral_mul_non_measurable₀'
theorem setLIntegral_withDensity_eq_setLIntegral_mul_non_measurable₀' (μ : Measure α) [SFinite μ]
    {f : α → ℝ≥0∞} (s : Set α) (hf : AEMeasurable f (μ.restrict s)) (g : α → ℝ≥0∞)
    (h'f : ∀ᵐ x ∂μ.restrict s, f x < ∞) :
    ∫⁻ a in s, g a ∂μ.withDensity f = ∫⁻ a in s, (f * g) a ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ] {f : α → ℝ≥0∞} (s : Set α) (hf : AEMeasurable f (μ.restrict s)) (g : α → ℝ≥0∞) (h'f : ∀ᵐ (x : α) ∂μ.restrict s, f x < ⊤)
Goal: ∫⁻ (a : α) in s, g a ∂μ.withDensity f = ∫⁻ (a : α) in s, (f * g) a ∂μ
-- MeasureTheory.withDensity_mul₀
theorem withDensity_mul₀ {μ : Measure α} {f g : α → ℝ≥0∞}
    (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
    μ.withDensity (f * g) = (μ.withDensity f).withDensity g 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
Goal: μ.withDensity (f * g) = (μ.withDensity f).withDensity g
-- MeasureTheory.withDensity_mul
theorem withDensity_mul (μ : Measure α) {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g) :
    μ.withDensity (f * g) = (μ.withDensity f).withDensity g 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g)
Goal: μ.withDensity (f * g) = (μ.withDensity f).withDensity g
-- MeasureTheory.withDensity_inv_same_le
lemma withDensity_inv_same_le {μ : Measure α} {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) :
    (μ.withDensity f).withDensity f⁻¹ ≤ μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : AEMeasurable f μ)
Goal: (μ.withDensity f).withDensity f⁻¹ ≤ μ
-- MeasureTheory.withDensity_inv_same₀
lemma withDensity_inv_same₀ {μ : Measure α} {f : α → ℝ≥0∞}
    (hf : AEMeasurable f μ) (hf_ne_zero : ∀ᵐ x ∂μ, f x ≠ 0) (hf_ne_top : ∀ᵐ x ∂μ, f x ≠ ∞) :
    (μ.withDensity f).withDensity (fun x ↦ (f x)⁻¹) = μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hf_ne_zero : ∀ᵐ (x : α) ∂μ, f x ≠ 0) (hf_ne_top : ∀ᵐ (x : α) ∂μ, f x ≠ ⊤)
Goal: ((μ.withDensity f).withDensity fun x => (f x)⁻¹) = μ
-- MeasureTheory.withDensity_inv_same
lemma withDensity_inv_same {μ : Measure α} {f : α → ℝ≥0∞}
    (hf : Measurable f) (hf_ne_zero : ∀ᵐ x ∂μ, f x ≠ 0) (hf_ne_top : ∀ᵐ x ∂μ, f x ≠ ∞) :
    (μ.withDensity f).withDensity (fun x ↦ (f x)⁻¹) = μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : Measurable f) (hf_ne_zero : ∀ᵐ (x : α) ∂μ, f x ≠ 0) (hf_ne_top : ∀ᵐ (x : α) ∂μ, f x ≠ ⊤)
Goal: ((μ.withDensity f).withDensity fun x => (f x)⁻¹) = μ
-- MeasureTheory.withDensity_absolutelyContinuous'
lemma withDensity_absolutelyContinuous' {μ : Measure α} {f : α → ℝ≥0∞}
    (hf : AEMeasurable f μ) (hf_ne_zero : ∀ᵐ x ∂μ, f x ≠ 0) :
    μ ≪ μ.withDensity f 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hf_ne_zero : ∀ᵐ (x : α) ∂μ, f x ≠ 0)
Goal: μ.AbsolutelyContinuous (μ.withDensity f)
-- MeasureTheory.withDensity_ae_eq
theorem withDensity_ae_eq {β : Type} {f g : α → β} {d : α → ℝ≥0∞}
    (hd : AEMeasurable d μ) (h_ae_nonneg : ∀ᵐ x ∂μ, d x ≠ 0) :
    f =ᵐ[μ.withDensity d] g ↔ f =ᵐ[μ] g 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type} {f g : α → β} {d : α → ℝ≥0∞} (hd : AEMeasurable d μ) (h_ae_nonneg : ∀ᵐ (x : α) ∂μ, d x ≠ 0)
Goal: f =ᵐ[μ.withDensity d] g ↔ f =ᵐ[μ] g
-- MeasureTheory.SigmaFinite.withDensity_of_ne_top
lemma SigmaFinite.withDensity_of_ne_top [SigmaFinite μ] {f : α → ℝ≥0∞}
    (hf_ne_top : ∀ᵐ x ∂μ, f x ≠ ∞) : SigmaFinite (μ.withDensity f) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] {f : α → ℝ≥0∞} (hf_ne_top : ∀ᵐ (x : α) ∂μ, f x ≠ ⊤)
Goal: MeasureTheory.SigmaFinite (μ.withDensity f)
-- MeasureTheory.SigmaFinite.withDensity_of_ne_top'
lemma SigmaFinite.withDensity_of_ne_top' [SigmaFinite μ] {f : α → ℝ≥0∞} (hf_ne_top : ∀ x, f x ≠ ∞) :
    SigmaFinite (μ.withDensity f) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] {f : α → ℝ≥0∞} (hf_ne_top : ∀ (x : α), f x ≠ ⊤)
Goal: MeasureTheory.SigmaFinite (μ.withDensity f)
-- MeasureTheory.exists_measurable_le_withDensity_eq
theorem exists_measurable_le_withDensity_eq [SFinite μ] (f : α → ℝ≥0∞) :
    ∃ g, Measurable g ∧ g ≤ f ∧ μ.withDensity g = μ.withDensity f 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ] (f : α → ℝ≥0∞)
Goal: ∃ g, Measurable g ∧ g ≤ f ∧ μ.withDensity g = μ.withDensity f
-- MeasureTheory.sFinite_withDensity_of_sigmaFinite_of_measurable
@[deprecated Measure.withDensity.instSFinite (since := "2024-07-14"), nolint unusedArguments]
lemma sFinite_withDensity_of_sigmaFinite_of_measurable (μ : Measure α) [SigmaFinite μ]
    {f : α → ℝ≥0∞} (_hf : Measurable f) :
    SFinite (μ.withDensity f) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] {f : α → ℝ≥0∞} (_hf : Measurable f)
Goal: MeasureTheory.SFinite (μ.withDensity f)
-- MeasureTheory.sFinite_withDensity_of_measurable
@[deprecated Measure.withDensity.instSFinite (since := "2024-07-14"), nolint unusedArguments]
lemma sFinite_withDensity_of_measurable (μ : Measure α) [SFinite μ]
    {f : α → ℝ≥0∞} (_hf : Measurable f) :
    SFinite (μ.withDensity f) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ] {f : α → ℝ≥0∞} (_hf : Measurable f)
Goal: MeasureTheory.SFinite (μ.withDensity f)
-- MeasureTheory.IsLocallyFiniteMeasure.withDensity_coe
lemma IsLocallyFiniteMeasure.withDensity_coe {f : α → ℝ≥0} (hf : Continuous f) :
    IsLocallyFiniteMeasure (μ.withDensity fun x ↦ f x) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace α] [OpensMeasurableSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] {f : α → ℝ≥0} (hf : Continuous f)
Goal: MeasureTheory.IsLocallyFiniteMeasure (μ.withDensity fun x => ↑(f x))