integrable_integral_norm_of_integrable_comp

English

The map from AEStronglyMeasurable f to the ℝ≥0∞-valued lintegral is continuous in the appropriate norm.

Русский

Пусть f похож на AEStronglyMeasurable; линеграля непрерывна по норме.

LaTeX

$$$$ \\text{continuous}(f \\mapsto \\mathrm{lintegral}\\_{x} F(\\int f)) $$$$

Lean4

theorem integrable_integral_norm_of_integrable_comp (h_int : Integrable f (κ ∘ₘ μ)) :
    Integrable (fun x ↦ ∫ y, ‖f y‖ ∂κ x) μ :=
  by
  rw [Measure.comp_eq_comp_const_apply, integrable_comp_iff h_int.1] at h_int
  exact h_int.2

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