mul_continuousOn — Mathlib

English

Let g be continuous on K and f ∈ Lp on A, with A ⊆ K and K compact; then the product g·f is in Lp on A.

Русский

Пусть g непрерывна на K, f ∈ Lp на A, A ⊆ K, K компакт; тогда произведение g·f ∈ Lp на A.

LaTeX

$$$ (g\cdot f) \in L^{p}(A,\mu) $$$

Lean4

theorem mul_continuousOn [T2Space X] (hg : IntegrableOn g K μ) (hg' : ContinuousOn g' K) (hK : IsCompact K) :
    IntegrableOn (fun x => g x * g' x) K μ :=
  hg.mul_continuousOn_of_subset hg' hK.measurableSet hK (Subset.refl _)

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