continuous_primitive — Mathlib

English

If a family of integrable functions has uniformly bounded integrals and converges pointwise, then the limit function is integrable and the limit of integrals equals the integral of the limit.

Русский

Если семейство интегрируемых функций сходится по точкам и интегралы ограничены, предел интегрируем и предел интегралов равен интегралу предела.

LaTeX

$$$\\int f = \\lim_{n\\to\\infty} \\int f_n$ under appropriate domination$$

Lean4

theorem continuous_primitive (h_int : ∀ a b, IntervalIntegrable f μ a b) (a : ℝ) :
    Continuous fun b => ∫ x in a..b, f x ∂μ :=
  by
  rw [continuous_iff_continuousAt]
  intro b₀
  obtain ⟨b₁, hb₁⟩ := exists_lt b₀
  obtain ⟨b₂, hb₂⟩ := exists_gt b₀
  apply ContinuousWithinAt.continuousAt _ (Icc_mem_nhds hb₁ hb₂)
  exact continuousWithinAt_primitive (measure_singleton b₀) (h_int _ _)

Похожие утверждения