File: Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean

full_name: MeasureTheory.Measure.withDensityᵥ
code: /-- Given a measure `μ` and an integrable function `f`, `μ.withDensityᵥ f` is
the vector measure which maps the set `s` to `∫ₛ f ∂μ`. -/
def Measure.withDensityᵥ {m : MeasurableSpace α} (μ : Measure α) (f : α → E) : VectorMeasure α E 
goal: MeasureTheory.VectorMeasure α E

full_name: MeasureTheory.withDensityᵥ_apply
code: theorem withDensityᵥ_apply (hf : Integrable f μ) {s : Set α} (hs : MeasurableSet s) :
    μ.withDensityᵥ f s = ∫ x in s, f x ∂μ 
goal: ↑(μ.withDensityᵥ f) s = ∫ (x : α) in s, f x ∂μ

full_name: MeasureTheory.withDensityᵥ_zero
code: @[simp]
theorem withDensityᵥ_zero : μ.withDensityᵥ (0 : α → E) = 0 
goal: μ.withDensityᵥ 0 = 0

full_name: MeasureTheory.withDensityᵥ_neg
code: @[simp]
theorem withDensityᵥ_neg : μ.withDensityᵥ (-f) = -μ.withDensityᵥ f 
goal: μ.withDensityᵥ (-f) = -μ.withDensityᵥ f

full_name: MeasureTheory.withDensityᵥ_neg'
code: theorem withDensityᵥ_neg' : (μ.withDensityᵥ fun x => -f x) = -μ.withDensityᵥ f 
goal: (μ.withDensityᵥ fun x => -f x) = -μ.withDensityᵥ f

full_name: MeasureTheory.withDensityᵥ_add
code: @[simp]
theorem withDensityᵥ_add (hf : Integrable f μ) (hg : Integrable g μ) :
    μ.withDensityᵥ (f + g) = μ.withDensityᵥ f + μ.withDensityᵥ g 
goal: μ.withDensityᵥ (f + g) = μ.withDensityᵥ f + μ.withDensityᵥ g

full_name: MeasureTheory.withDensityᵥ_add'
code: theorem withDensityᵥ_add' (hf : Integrable f μ) (hg : Integrable g μ) :
    (μ.withDensityᵥ fun x => f x + g x) = μ.withDensityᵥ f + μ.withDensityᵥ g 
goal: (μ.withDensityᵥ fun x => f x + g x) = μ.withDensityᵥ f + μ.withDensityᵥ g

full_name: MeasureTheory.withDensityᵥ_sub
code: @[simp]
theorem withDensityᵥ_sub (hf : Integrable f μ) (hg : Integrable g μ) :
    μ.withDensityᵥ (f - g) = μ.withDensityᵥ f - μ.withDensityᵥ g 
goal: μ.withDensityᵥ (f - g) = μ.withDensityᵥ f - μ.withDensityᵥ g

full_name: MeasureTheory.withDensityᵥ_sub'
code: theorem withDensityᵥ_sub' (hf : Integrable f μ) (hg : Integrable g μ) :
    (μ.withDensityᵥ fun x => f x - g x) = μ.withDensityᵥ f - μ.withDensityᵥ g 
goal: (μ.withDensityᵥ fun x => f x - g x) = μ.withDensityᵥ f - μ.withDensityᵥ g

full_name: MeasureTheory.withDensityᵥ_smul
code: @[simp]
theorem withDensityᵥ_smul {𝕜 : Type*} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E]
    [SMulCommClass ℝ 𝕜 E] (f : α → E) (r : 𝕜) : μ.withDensityᵥ (r • f) = r • μ.withDensityᵥ f 
goal: μ.withDensityᵥ (r • f) = r • μ.withDensityᵥ f

full_name: MeasureTheory.withDensityᵥ_smul'
code: theorem withDensityᵥ_smul' {𝕜 : Type*} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E]
    [SMulCommClass ℝ 𝕜 E] (f : α → E) (r : 𝕜) :
    (μ.withDensityᵥ fun x => r • f x) = r • μ.withDensityᵥ f 
goal: (μ.withDensityᵥ fun x => r • f x) = r • μ.withDensityᵥ f

full_name: MeasureTheory.withDensityᵥ_smul_eq_withDensityᵥ_withDensity
code: theorem withDensityᵥ_smul_eq_withDensityᵥ_withDensity {f : α → ℝ≥0} {g : α → E}
    (hf : AEMeasurable f μ) (hfg : Integrable (f • g) μ) :
    μ.withDensityᵥ (f • g) = (μ.withDensity (fun x ↦ f x)).withDensityᵥ g 
goal: μ.withDensityᵥ (f • g) = (μ.withDensity fun x => ↑(f x)).withDensityᵥ g

full_name: MeasureTheory.withDensityᵥ_smul_eq_withDensityᵥ_withDensity'
code: theorem withDensityᵥ_smul_eq_withDensityᵥ_withDensity' {f : α → ℝ≥0∞} {g : α → E}
    (hf : AEMeasurable f μ) (hflt : ∀ᵐ x ∂μ, f x < ∞)
    (hfg : Integrable (fun x ↦ (f x).toReal • g x) μ) :
    μ.withDensityᵥ (fun x ↦ (f x).toReal • g x) = (μ.withDensity f).withDensityᵥ g 
goal: (μ.withDensityᵥ fun x => (f x).toReal • g x) = (μ.withDensity f).withDensityᵥ g

full_name: MeasureTheory.Measure.withDensityᵥ_absolutelyContinuous
code: theorem Measure.withDensityᵥ_absolutelyContinuous (μ : Measure α) (f : α → ℝ) :
    μ.withDensityᵥ f ≪ᵥ μ.toENNRealVectorMeasure 
goal: μ.withDensityᵥ f ≪ᵥ μ.toENNRealVectorMeasure

full_name: MeasureTheory.Integrable.ae_eq_of_withDensityᵥ_eq
code: /-- Having the same density implies the underlying functions are equal almost everywhere. -/
theorem Integrable.ae_eq_of_withDensityᵥ_eq {f g : α → E} (hf : Integrable f μ)
    (hg : Integrable g μ) (hfg : μ.withDensityᵥ f = μ.withDensityᵥ g) : f =ᵐ[μ] g 
goal: f =ᵐ[μ] g

full_name: MeasureTheory.WithDensityᵥEq.congr_ae
code: theorem WithDensityᵥEq.congr_ae {f g : α → E} (h : f =ᵐ[μ] g) :
    μ.withDensityᵥ f = μ.withDensityᵥ g 
goal: μ.withDensityᵥ f = μ.withDensityᵥ g

full_name: MeasureTheory.Integrable.withDensityᵥ_eq_iff
code: theorem Integrable.withDensityᵥ_eq_iff {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) :
    μ.withDensityᵥ f = μ.withDensityᵥ g ↔ f =ᵐ[μ] g 
goal: μ.withDensityᵥ f = μ.withDensityᵥ g ↔ f =ᵐ[μ] g

full_name: MeasureTheory.withDensityᵥ_toReal
code: theorem withDensityᵥ_toReal {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ) (hf : (∫⁻ x, f x ∂μ) ≠ ∞) :
    (μ.withDensityᵥ fun x => (f x).toReal) =
      @toSignedMeasure α _ (μ.withDensity f) (isFiniteMeasure_withDensity hf) 
goal: (μ.withDensityᵥ fun x => (f x).toReal) = (μ.withDensity f).toSignedMeasure

full_name: MeasureTheory.withDensityᵥ_eq_withDensity_pos_part_sub_withDensity_neg_part
code: theorem withDensityᵥ_eq_withDensity_pos_part_sub_withDensity_neg_part {f : α → ℝ}
    (hfi : Integrable f μ) :
    μ.withDensityᵥ f =
      @toSignedMeasure α _ (μ.withDensity fun x => ENNReal.ofReal <| f x)
          (isFiniteMeasure_withDensity_ofReal hfi.2) -
        @toSignedMeasure α _ (μ.withDensity fun x => ENNReal.ofReal <| -f x)
          (isFiniteMeasure_withDensity_ofReal hfi.neg.2) 
goal: μ.withDensityᵥ f =
  (μ.withDensity fun x => ENNReal.ofReal (f x)).toSignedMeasure -
    (μ.withDensity fun x => ENNReal.ofReal (-f x)).toSignedMeasure

full_name: MeasureTheory.Integrable.withDensityᵥ_trim_eq_integral
code: theorem Integrable.withDensityᵥ_trim_eq_integral {m m0 : MeasurableSpace α} {μ : Measure α}
    (hm : m ≤ m0) {f : α → ℝ} (hf : Integrable f μ) {i : Set α} (hi : MeasurableSet[m] i) :
    (μ.withDensityᵥ f).trim hm i = ∫ x in i, f x ∂μ 
goal: ↑((μ.withDensityᵥ f).trim hm) i = ∫ (x : α) in i, f x ∂μ

full_name: MeasureTheory.Integrable.withDensityᵥ_trim_absolutelyContinuous
code: theorem Integrable.withDensityᵥ_trim_absolutelyContinuous {m m0 : MeasurableSpace α} {μ : Measure α}
    (hm : m ≤ m0) (hfi : Integrable f μ) :
    (μ.withDensityᵥ f).trim hm ≪ᵥ (μ.trim hm).toENNRealVectorMeasure 
goal: (μ.withDensityᵥ f).trim hm ≪ᵥ (μ.trim hm).toENNRealVectorMeasure