apply_continuousLinearMap — Mathlib

English

Equivalence for a NNReal density function f: AEStronglyMeasurable g w.r.t μ.withDensity(f) iff scaled AEStronglyMeasurable under μ.

Русский

Эквивалентность для плотности NNReal: AEStronglyMeasurable g по мере μ.withDensity f ⇔ AEStronglyMeasurable (f•g) по μ.

LaTeX

$$$$\\text{AEStronglyMeasurable}(g, \\mu.withDensity f) \\iff \\text{AEStronglyMeasurable}( (f) \\cdot g, \\mu ).$$$$

Lean4

@[measurability]
theorem apply_continuousLinearMap {φ : α → F →L[𝕜] E} (hφ : AEStronglyMeasurable φ μ) (v : F) :
    AEStronglyMeasurable (fun a => φ a v) μ :=
  (ContinuousLinearMap.apply 𝕜 E v).continuous.comp_aestronglyMeasurable hφ

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