Derivatives of integrals depending on parameters

A parametric integral is a function with shape f = fun x : H ↦ ∫ a : α, F x a ∂μ for some F : H → α → E, where H and E are normed spaces and α is a measured space with measure μ.

We already know from continuous_of_dominated in Mathlib/MeasureTheory/Integral/Bochner.lean how to guarantee that f is continuous using the dominated convergence theorem. In this file, we want to express the derivative of f as the integral of the derivative of F with respect to x.

Main results

As explained above, all results express the derivative of a parametric integral as the integral of a derivative. The variations come from the assumptions and from the different ways of expressing derivative, especially Fréchet derivatives vs elementary derivative of function of one real variable.

• hasFDerivAt_integral_of_dominated_loc_of_lip: this version assumes that

  • F x is ae-measurable for x near x₀,
  • F x₀ is integrable,
  • fun x ↦ F x a has derivative F' a : H →L[ℝ] E at x₀ which is ae-measurable,
  • fun x ↦ F x a is locally Lipschitz near x₀ for almost every a, with a Lipschitz bound which is integrable with respect to a.

  A subtle point is that the "near x₀" in the last condition has to be uniform in a. This is controlled by a positive number ε.

• hasFDerivAt_integral_of_dominated_of_fderiv_le: this version assumes fun x ↦ F x a has derivative F' x a for x near x₀ and F' x is bounded by an integrable function independent from x near x₀.

hasDerivAt_integral_of_dominated_loc_of_lip and hasDerivAt_integral_of_dominated_loc_of_deriv_le are versions of the above two results that assume H = ℝ or H = ℂ and use the high-school derivative deriv instead of Fréchet derivative fderiv.

We also provide versions of these theorems for set integrals.

Tags

integral, derivative

theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedSpace 𝕜 E] {H : Type u_4} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {F : H → α → E} {x₀ : H} {bound : α → ℝ} {ε : ℝ} {F' : α → H →L[𝕜] E} (ε_pos : 0 < ε) (hF_meas : ∀ x ∈ Metric.ball x₀ ε, MeasureTheory.AEStronglyMeasurable (F x) μ) (hF_int : MeasureTheory.Integrable (F x₀) μ) (hF'_meas : MeasureTheory.AEStronglyMeasurable F' μ) (h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ Metric.ball x₀ ε, ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖) (bound_integrable : MeasureTheory.Integrable bound μ) (h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun (x : H) => F x a) (F' a) x₀) : MeasureTheory.Integrable F' μ ∧ HasFDerivAt (fun (x : H) => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀

Differentiation under integral of x ↦ ∫ F x a at a given point x₀, assuming F x₀ is integrable, ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖ for x in a ball around x₀ for ae a with integrable Lipschitz bound bound (with a ball radius independent of a), and F x is ae-measurable for x in the same ball. See hasFDerivAt_integral_of_dominated_loc_of_lip for a slightly less general but usually more useful version.

theorem hasFDerivAt_integral_of_dominated_loc_of_lip {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedSpace 𝕜 E] {H : Type u_4} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {F : H → α → E} {x₀ : H} {bound : α → ℝ} {ε : ℝ} {F' : α → H →L[𝕜] E} (ε_pos : 0 < ε) (hF_meas : ∀ᶠ (x : H) in nhds x₀, MeasureTheory.AEStronglyMeasurable (F x) μ) (hF_int : MeasureTheory.Integrable (F x₀) μ) (hF'_meas : MeasureTheory.AEStronglyMeasurable F' μ) (h_lip : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (fun (x : H) => F x a) (Metric.ball x₀ ε)) (bound_integrable : MeasureTheory.Integrable bound μ) (h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun (x : H) => F x a) (F' a) x₀) : MeasureTheory.Integrable F' μ ∧ HasFDerivAt (fun (x : H) => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀

Differentiation under integral of x ↦ ∫ F x a at a given point x₀, assuming F x₀ is integrable, x ↦ F x a is locally Lipschitz on a ball around x₀ for ae a (with a ball radius independent of a) with integrable Lipschitz bound, and F x is ae-measurable for x in a possibly smaller neighborhood of x₀.

theorem hasFDerivAt_integral_of_dominated_loc_of_lip_interval {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_4} [NormedAddCommGroup H] {x₀ : H} {ε : ℝ} [NormedSpace ℝ H] {μ : MeasureTheory.Measure ℝ} {F : H → ℝ → E} {F' : ℝ → H →L[ℝ] E} {a : ℝ} {b : ℝ} {bound : ℝ → ℝ} (ε_pos : 0 < ε) (hF_meas : ∀ᶠ (x : H) in nhds x₀, MeasureTheory.AEStronglyMeasurable (F x) (μ.restrict (Ι a b))) (hF_int : IntervalIntegrable (F x₀) μ a b) (hF'_meas : MeasureTheory.AEStronglyMeasurable F' (μ.restrict (Ι a b))) (h_lip : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), LipschitzOnWith (Real.nnabs (bound t)) (fun (x : H) => F x t) (Metric.ball x₀ ε)) (bound_integrable : IntervalIntegrable bound μ a b) (h_diff : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), HasFDerivAt (fun (x : H) => F x t) (F' t) x₀) : IntervalIntegrable F' μ a b ∧ HasFDerivAt (fun (x : H) => ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' t ∂μ) x₀

Differentiation under integral of x ↦ ∫ x in a..b, F x t at a given point x₀ ∈ (a,b), assuming F x₀ is integrable on (a,b), that x ↦ F x t is Lipschitz on a ball around x₀ for almost every t (with a ball radius independent of t) with integrable Lipschitz bound, and F x is a.e.-measurable for x in a possibly smaller neighborhood of x₀.

theorem hasFDerivAt_integral_of_dominated_of_fderiv_le {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedSpace 𝕜 E] {H : Type u_4} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {F : H → α → E} {x₀ : H} {bound : α → ℝ} {ε : ℝ} {F' : H → α → H →L[𝕜] E} (ε_pos : 0 < ε) (hF_meas : ∀ᶠ (x : H) in nhds x₀, MeasureTheory.AEStronglyMeasurable (F x) μ) (hF_int : MeasureTheory.Integrable (F x₀) μ) (hF'_meas : MeasureTheory.AEStronglyMeasurable (F' x₀) μ) (h_bound : ∀ᵐ (a : α) ∂μ, ∀ x ∈ Metric.ball x₀ ε, ‖F' x a‖ ≤ bound a) (bound_integrable : MeasureTheory.Integrable bound μ) (h_diff : ∀ᵐ (a : α) ∂μ, ∀ x ∈ Metric.ball x₀ ε, HasFDerivAt (fun (x : H) => F x a) (F' x a) x) : HasFDerivAt (fun (x : H) => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' x₀ a ∂μ) x₀

Differentiation under integral of x ↦ ∫ F x a at a given point x₀, assuming F x₀ is integrable, x ↦ F x a is differentiable on a ball around x₀ for ae a with derivative norm uniformly bounded by an integrable function (the ball radius is independent of a), and F x is ae-measurable for x in a possibly smaller neighborhood of x₀.

theorem hasFDerivAt_integral_of_dominated_of_fderiv_le'' {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_4} [NormedAddCommGroup H] {x₀ : H} {ε : ℝ} [NormedSpace ℝ H] {μ : MeasureTheory.Measure ℝ} {F : H → ℝ → E} {F' : H → ℝ → H →L[ℝ] E} {a : ℝ} {b : ℝ} {bound : ℝ → ℝ} (ε_pos : 0 < ε) (hF_meas : ∀ᶠ (x : H) in nhds x₀, MeasureTheory.AEStronglyMeasurable (F x) (μ.restrict (Ι a b))) (hF_int : IntervalIntegrable (F x₀) μ a b) (hF'_meas : MeasureTheory.AEStronglyMeasurable (F' x₀) (μ.restrict (Ι a b))) (h_bound : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ∀ x ∈ Metric.ball x₀ ε, ‖F' x t‖ ≤ bound t) (bound_integrable : IntervalIntegrable bound μ a b) (h_diff : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ∀ x ∈ Metric.ball x₀ ε, HasFDerivAt (fun (x : H) => F x t) (F' x t) x) : HasFDerivAt (fun (x : H) => ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' x₀ t ∂μ) x₀

Differentiation under integral of x ↦ ∫ x in a..b, F x a at a given point x₀, assuming F x₀ is integrable on (a,b), x ↦ F x a is differentiable on a ball around x₀ for ae a with derivative norm uniformly bounded by an integrable function (the ball radius is independent of a), and F x is ae-measurable for x in a possibly smaller neighborhood of x₀.

theorem hasDerivAt_integral_of_dominated_loc_of_lip {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedSpace 𝕜 E] {bound : α → ℝ} {ε : ℝ} {F : 𝕜 → α → E} {x₀ : 𝕜} {F' : α → E} (ε_pos : 0 < ε) (hF_meas : ∀ᶠ (x : 𝕜) in nhds x₀, MeasureTheory.AEStronglyMeasurable (F x) μ) (hF_int : MeasureTheory.Integrable (F x₀) μ) (hF'_meas : MeasureTheory.AEStronglyMeasurable F' μ) (h_lipsch : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (fun (x : 𝕜) => F x a) (Metric.ball x₀ ε)) (bound_integrable : MeasureTheory.Integrable bound μ) (h_diff : ∀ᵐ (a : α) ∂μ, HasDerivAt (fun (x : 𝕜) => F x a) (F' a) x₀) : MeasureTheory.Integrable F' μ ∧ HasDerivAt (fun (x : 𝕜) => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀

Derivative under integral of x ↦ ∫ F x a at a given point x₀ : 𝕜, 𝕜 = ℝ or 𝕜 = ℂ, assuming F x₀ is integrable, x ↦ F x a is locally Lipschitz on a ball around x₀ for ae a (with ball radius independent of a) with integrable Lipschitz bound, and F x is ae-measurable for x in a possibly smaller neighborhood of x₀.

theorem hasDerivAt_integral_of_dominated_loc_of_deriv_le {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedSpace 𝕜 E] {bound : α → ℝ} {ε : ℝ} {F : 𝕜 → α → E} {x₀ : 𝕜} (ε_pos : 0 < ε) (hF_meas : ∀ᶠ (x : 𝕜) in nhds x₀, MeasureTheory.AEStronglyMeasurable (F x) μ) (hF_int : MeasureTheory.Integrable (F x₀) μ) {F' : 𝕜 → α → E} (hF'_meas : MeasureTheory.AEStronglyMeasurable (F' x₀) μ) (h_bound : ∀ᵐ (a : α) ∂μ, ∀ x ∈ Metric.ball x₀ ε, ‖F' x a‖ ≤ bound a) (bound_integrable : MeasureTheory.Integrable bound μ) (h_diff : ∀ᵐ (a : α) ∂μ, ∀ x ∈ Metric.ball x₀ ε, HasDerivAt (fun (x : 𝕜) => F x a) (F' x a) x) : MeasureTheory.Integrable (F' x₀) μ ∧ HasDerivAt (fun (n : 𝕜) => ∫ (a : α), F n a ∂μ) (∫ (a : α), F' x₀ a ∂μ) x₀

Derivative under integral of x ↦ ∫ F x a at a given point x₀ : ℝ, assuming F x₀ is integrable, x ↦ F x a is differentiable on an interval around x₀ for ae a (with interval radius independent of a) with derivative uniformly bounded by an integrable function, and F x is ae-measurable for x in a possibly smaller neighborhood of x₀.