mathlib
2f40f351 - feat(measure_theory): continuity of primitives (#7864)

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4 years ago
feat(measure_theory): continuity of primitives (#7864) From the sphere eversion project This proves some continuity of interval integrals with respect to parameters and continuity of primitives of measurable functions. The statements are a bit abstract, but they allow to have: ```lean example {f : ℝ → E} (h_int : integrable f) (a : ℝ) : continuous (λ b, ∫ x in a .. b, f x ∂ volume) := h_int.continuous_primitive a ``` under the usual assumptions on `E`: `normed_group E`, `second_countable_topology E`, `normed_space ℝ E` `complete_space E`, `measurable_space E`, `borel_space E`, say `E = ℝ` for instance. Of course global integrability is not needed, assuming integrability on all finite length intervals is enough: ```lean example {f : ℝ → E} (h_int : ∀ a b : ℝ, interval_integrable f volume a b) (a : ℝ) : continuous (λ b, ∫ x in a .. b, f x ∂ volume) := continuous_primitive h_int a ```
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