Derivatives of interval integrals depending on parameters #

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In this file we restate theorems about derivatives of integrals depending on parameters for interval integrals.

theorem interval_integral.has_fderiv_at_integral_of_dominated_loc_of_lip {𝕜 : Type u_1} [is_R_or_C 𝕜] {μ : measure_theory.measure ℝ} {E : Type u_2} [normed_add_comm_group E] [normed_space ℝ E] [normed_space 𝕜 E] [complete_space E] {H : Type u_3} [normed_add_comm_group H] [normed_space 𝕜 H] {a b ε : ℝ} {bound : ℝ → ℝ} {F : H → ℝ → E} {F' : ℝ → (H →L[𝕜] E)} {x₀ : H} (ε_pos : 0 < ε) (hF_meas : ∀ᶠ (x : H) in nhds x₀, measure_theory.ae_strongly_measurable (F x) (μ.restrict (set.uIoc a b))) (hF_int : interval_integrable (F x₀) μ a b) (hF'_meas : measure_theory.ae_strongly_measurable F' (μ.restrict (set.uIoc a b))) (h_lip : ∀ᵐ (t : ℝ) ∂μ, t ∈ set.uIoc a b → lipschitz_on_with (⇑real.nnabs (bound t)) (λ (x : H), F x t) (metric.ball x₀ ε)) (bound_integrable : interval_integrable bound μ a b) (h_diff : ∀ᵐ (t : ℝ) ∂μ, t ∈ set.uIoc a b → has_fderiv_at (λ (x : H), F x t) (F' t) x₀) :

interval_integrable F' μ a b ∧ has_fderiv_at (λ (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 ↦ ∫ t in a..b, F x t 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.

source

theorem interval_integral.has_fderiv_at_integral_of_dominated_of_fderiv_le {𝕜 : Type u_1} [is_R_or_C 𝕜] {μ : measure_theory.measure ℝ} {E : Type u_2} [normed_add_comm_group E] [normed_space ℝ E] [normed_space 𝕜 E] [complete_space E] {H : Type u_3} [normed_add_comm_group H] [normed_space 𝕜 H] {a b ε : ℝ} {bound : ℝ → ℝ} {F : H → ℝ → E} {F' : H → ℝ → (H →L[𝕜] E)} {x₀ : H} (ε_pos : 0 < ε) (hF_meas : ∀ᶠ (x : H) in nhds x₀, measure_theory.ae_strongly_measurable (F x) (μ.restrict (set.uIoc a b))) (hF_int : interval_integrable (F x₀) μ a b) (hF'_meas : measure_theory.ae_strongly_measurable (F' x₀) (μ.restrict (set.uIoc a b))) (h_bound : ∀ᵐ (t : ℝ) ∂μ, t ∈ set.uIoc a b → ∀ (x : H), x ∈ metric.ball x₀ ε → ‖F' x t‖ ≤ bound t) (bound_integrable : interval_integrable bound μ a b) (h_diff : ∀ᵐ (t : ℝ) ∂μ, t ∈ set.uIoc a b → ∀ (x : H), x ∈ metric.ball x₀ ε → has_fderiv_at (λ (x : H), F x t) (F' x t) x) :

has_fderiv_at (λ (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 ↦ ∫ 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.

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theorem interval_integral.has_deriv_at_integral_of_dominated_loc_of_lip {𝕜 : Type u_1} [is_R_or_C 𝕜] {μ : measure_theory.measure ℝ} {E : Type u_2} [normed_add_comm_group E] [normed_space ℝ E] [normed_space 𝕜 E] [complete_space E] {a b ε : ℝ} {bound : ℝ → ℝ} {F : 𝕜 → ℝ → E} {F' : ℝ → E} {x₀ : 𝕜} (ε_pos : 0 < ε) (hF_meas : ∀ᶠ (x : 𝕜) in nhds x₀, measure_theory.ae_strongly_measurable (F x) (μ.restrict (set.uIoc a b))) (hF_int : interval_integrable (F x₀) μ a b) (hF'_meas : measure_theory.ae_strongly_measurable F' (μ.restrict (set.uIoc a b))) (h_lipsch : ∀ᵐ (t : ℝ) ∂μ, t ∈ set.uIoc a b → lipschitz_on_with (⇑real.nnabs (bound t)) (λ (x : 𝕜), F x t) (metric.ball x₀ ε)) (bound_integrable : interval_integrable bound μ a b) (h_diff : ∀ᵐ (t : ℝ) ∂μ, t ∈ set.uIoc a b → has_deriv_at (λ (x : 𝕜), F x t) (F' t) x₀) :

interval_integrable F' μ a b ∧ has_deriv_at (λ (x : 𝕜), ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' t ∂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.

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theorem interval_integral.has_deriv_at_integral_of_dominated_loc_of_deriv_le {𝕜 : Type u_1} [is_R_or_C 𝕜] {μ : measure_theory.measure ℝ} {E : Type u_2} [normed_add_comm_group E] [normed_space ℝ E] [normed_space 𝕜 E] [complete_space E] {a b ε : ℝ} {bound : ℝ → ℝ} {F F' : 𝕜 → ℝ → E} {x₀ : 𝕜} (ε_pos : 0 < ε) (hF_meas : ∀ᶠ (x : 𝕜) in nhds x₀, measure_theory.ae_strongly_measurable (F x) (μ.restrict (set.uIoc a b))) (hF_int : interval_integrable (F x₀) μ a b) (hF'_meas : measure_theory.ae_strongly_measurable (F' x₀) (μ.restrict (set.uIoc a b))) (h_bound : ∀ᵐ (t : ℝ) ∂μ, t ∈ set.uIoc a b → ∀ (x : 𝕜), x ∈ metric.ball x₀ ε → ‖F' x t‖ ≤ bound t) (bound_integrable : interval_integrable bound μ a b) (h_diff : ∀ᵐ (t : ℝ) ∂μ, t ∈ set.uIoc a b → ∀ (x : 𝕜), x ∈ metric.ball x₀ ε → has_deriv_at (λ (x : 𝕜), F x t) (F' x t) x) :

interval_integrable (F' x₀) μ a b ∧ has_deriv_at (λ (x : 𝕜), ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' x₀ t ∂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 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.