File: Mathlib/Probability/Integration.lean

full_name: ProbabilityTheory.lintegral_mul_indicator_eq_lintegral_mul_lintegral_indicator
code: /-- If a random variable `f` in `ℝ≥0∞` is independent of an event `T`, then if you restrict the
  random variable to `T`, then `E[f * indicator T c 0]=E[f] * E[indicator T c 0]`. It is useful for
  `lintegral_mul_eq_lintegral_mul_lintegral_of_independent_measurableSpace`. -/
theorem lintegral_mul_indicator_eq_lintegral_mul_lintegral_indicator {Mf mΩ : MeasurableSpace Ω}
    {μ : Measure Ω} (hMf : Mf ≤ mΩ) (c : ℝ≥0∞) {T : Set Ω} (h_meas_T : MeasurableSet T)
    (h_ind : IndepSets {s | MeasurableSet[Mf] s} {T} μ) (h_meas_f : Measurable[Mf] f) :
    (∫⁻ ω, f ω * T.indicator (fun _ => c) ω ∂μ) =
      (∫⁻ ω, f ω ∂μ) * ∫⁻ ω, T.indicator (fun _ => c) ω ∂μ 
goal: ∫⁻ (ω : Ω), f ω * T.indicator (fun x => c) ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), T.indicator (fun x => c) ω ∂μ

full_name: ProbabilityTheory.lintegral_mul_eq_lintegral_mul_lintegral_of_independent_measurableSpace
code: /-- If `f` and `g` are independent random variables with values in `ℝ≥0∞`,
   then `E[f * g] = E[f] * E[g]`. However, instead of directly using the independence
   of the random variables, it uses the independence of measurable spaces for the
   domains of `f` and `g`. This is similar to the sigma-algebra approach to
   independence. See `lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun` for
   a more common variant of the product of independent variables. -/
theorem lintegral_mul_eq_lintegral_mul_lintegral_of_independent_measurableSpace
    {Mf Mg mΩ : MeasurableSpace Ω} {μ : Measure Ω} (hMf : Mf ≤ mΩ) (hMg : Mg ≤ mΩ)
    (h_ind : Indep Mf Mg μ) (h_meas_f : Measurable[Mf] f) (h_meas_g : Measurable[Mg] g) :
    ∫⁻ ω, f ω * g ω ∂μ = (∫⁻ ω, f ω ∂μ) * ∫⁻ ω, g ω ∂μ 
goal: ∫⁻ (ω : Ω), f ω * g ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), g ω ∂μ

full_name: ProbabilityTheory.lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun
code: /-- If `f` and `g` are independent random variables with values in `ℝ≥0∞`,
   then `E[f * g] = E[f] * E[g]`. -/
theorem lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun (h_meas_f : Measurable f)
    (h_meas_g : Measurable g) (h_indep_fun : IndepFun f g μ) :
    (∫⁻ ω, (f * g) ω ∂μ) = (∫⁻ ω, f ω ∂μ) * ∫⁻ ω, g ω ∂μ 
goal: ∫⁻ (ω : Ω), (f * g) ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), g ω ∂μ

full_name: ProbabilityTheory.lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun'
code: /-- If `f` and `g` with values in `ℝ≥0∞` are independent and almost everywhere measurable,
   then `E[f * g] = E[f] * E[g]` (slightly generalizing
   `lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun`). -/
theorem lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun' (h_meas_f : AEMeasurable f μ)
    (h_meas_g : AEMeasurable g μ) (h_indep_fun : IndepFun f g μ) :
    (∫⁻ ω, (f * g) ω ∂μ) = (∫⁻ ω, f ω ∂μ) * ∫⁻ ω, g ω ∂μ 
goal: ∫⁻ (ω : Ω), (f * g) ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), g ω ∂μ

full_name: ProbabilityTheory.lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun''
code: theorem lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun'' (h_meas_f : AEMeasurable f μ)
    (h_meas_g : AEMeasurable g μ) (h_indep_fun : IndepFun f g μ) :
    ∫⁻ ω, f ω * g ω ∂μ = (∫⁻ ω, f ω ∂μ) * ∫⁻ ω, g ω ∂μ 
goal: ∫⁻ (ω : Ω), f ω * g ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), g ω ∂μ

full_name: ProbabilityTheory.IndepFun.integrable_mul
code: /-- The product of two independent, integrable, real-valued random variables is integrable. -/
theorem IndepFun.integrable_mul {β : Type*} [MeasurableSpace β] {X Y : Ω → β}
    [NormedDivisionRing β] [BorelSpace β] (hXY : IndepFun X Y μ) (hX : Integrable X μ)
    (hY : Integrable Y μ) : Integrable (X * Y) μ 
goal: MeasureTheory.Integrable (X * Y) μ

full_name: ProbabilityTheory.IndepFun.integrable_left_of_integrable_mul
code: /-- If the product of two independent real-valued random variables is integrable and
the second one is not almost everywhere zero, then the first one is integrable. -/
theorem IndepFun.integrable_left_of_integrable_mul {β : Type*} [MeasurableSpace β] {X Y : Ω → β}
    [NormedDivisionRing β] [BorelSpace β] (hXY : IndepFun X Y μ) (h'XY : Integrable (X * Y) μ)
    (hX : AEStronglyMeasurable X μ) (hY : AEStronglyMeasurable Y μ) (h'Y : ¬Y =ᵐ[μ] 0) :
    Integrable X μ 
goal: MeasureTheory.Integrable X μ

full_name: ProbabilityTheory.IndepFun.integrable_right_of_integrable_mul
code: /-- If the product of two independent real-valued random variables is integrable and the
first one is not almost everywhere zero, then the second one is integrable. -/
theorem IndepFun.integrable_right_of_integrable_mul {β : Type*} [MeasurableSpace β] {X Y : Ω → β}
    [NormedDivisionRing β] [BorelSpace β] (hXY : IndepFun X Y μ) (h'XY : Integrable (X * Y) μ)
    (hX : AEStronglyMeasurable X μ) (hY : AEStronglyMeasurable Y μ) (h'X : ¬X =ᵐ[μ] 0) :
    Integrable Y μ 
goal: MeasureTheory.Integrable Y μ

full_name: ProbabilityTheory.IndepFun.integral_mul_of_nonneg
code: /-- The (Bochner) integral of the product of two independent, nonnegative random
  variables is the product of their integrals. The proof is just plumbing around
  `lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun'`. -/
theorem IndepFun.integral_mul_of_nonneg (hXY : IndepFun X Y μ) (hXp : 0 ≤ X) (hYp : 0 ≤ Y)
    (hXm : AEMeasurable X μ) (hYm : AEMeasurable Y μ) :
    integral μ (X * Y) = integral μ X * integral μ Y 
goal: MeasureTheory.integral μ (X * Y) = MeasureTheory.integral μ X * MeasureTheory.integral μ Y

full_name: ProbabilityTheory.IndepFun.integral_mul_of_integrable
code: /-- The (Bochner) integral of the product of two independent, integrable random
  variables is the product of their integrals. The proof is pedestrian decomposition
  into their positive and negative parts in order to apply `IndepFun.integral_mul_of_nonneg`
  four times. -/
theorem IndepFun.integral_mul_of_integrable (hXY : IndepFun X Y μ) (hX : Integrable X μ)
    (hY : Integrable Y μ) : integral μ (X * Y) = integral μ X * integral μ Y 
goal: MeasureTheory.integral μ (X * Y) = MeasureTheory.integral μ X * MeasureTheory.integral μ Y

full_name: ProbabilityTheory.IndepFun.integral_mul
code: /-- The (Bochner) integral of the product of two independent random
  variables is the product of their integrals. -/
theorem IndepFun.integral_mul (hXY : IndepFun X Y μ) (hX : AEStronglyMeasurable X μ)
    (hY : AEStronglyMeasurable Y μ) : integral μ (X * Y) = integral μ X * integral μ Y 
goal: MeasureTheory.integral μ (X * Y) = MeasureTheory.integral μ X * MeasureTheory.integral μ Y

full_name: ProbabilityTheory.IndepFun.integral_mul'
code: theorem IndepFun.integral_mul' (hXY : IndepFun X Y μ) (hX : AEStronglyMeasurable X μ)
    (hY : AEStronglyMeasurable Y μ) :
    (integral μ fun ω => X ω * Y ω) = integral μ X * integral μ Y 
goal: ∫ (ω : Ω), X ω * Y ω ∂μ = MeasureTheory.integral μ X * MeasureTheory.integral μ Y

full_name: ProbabilityTheory.indepFun_iff_integral_comp_mul
code: /-- Independence of functions `f` and `g` into arbitrary types is characterized by the relation
  `E[(φ ∘ f) * (ψ ∘ g)] = E[φ ∘ f] * E[ψ ∘ g]` for all measurable `φ` and `ψ` with values in `ℝ`
  satisfying appropriate integrability conditions. -/
theorem indepFun_iff_integral_comp_mul [IsFiniteMeasure μ] {β β' : Type*} {mβ : MeasurableSpace β}
    {mβ' : MeasurableSpace β'} {f : Ω → β} {g : Ω → β'} {hfm : Measurable f} {hgm : Measurable g} :
    IndepFun f g μ ↔ ∀ {φ : β → ℝ} {ψ : β' → ℝ}, Measurable φ → Measurable ψ →
      Integrable (φ ∘ f) μ → Integrable (ψ ∘ g) μ →
        integral μ (φ ∘ f * ψ ∘ g) = integral μ (φ ∘ f) * integral μ (ψ ∘ g) 
goal: ProbabilityTheory.IndepFun f g μ ↔
  ∀ {φ : β → ℝ} {ψ : β' → ℝ},
    Measurable φ →
      Measurable ψ →
        MeasureTheory.Integrable (φ ∘ f) μ →
          MeasureTheory.Integrable (ψ ∘ g) μ →
            MeasureTheory.integral μ (φ ∘ f * ψ ∘ g) =
              MeasureTheory.integral μ (φ ∘ f) * MeasureTheory.integral μ (ψ ∘ g)