comp_fst — Mathlib

English

If f is measurable on α×β, then the function x ↦ f(x) depends measurably on the first coordinate when viewed on μ×ν.

Русский

Если f измерима на α×β, то функция по первой координате измерима при возведении.

LaTeX

$$$$\\text{AEStronglyMeasurable}\\left( x\\mapsto f(x)^{\\text{fst}}\\right)$$$$

Lean4

theorem comp_fst {γ} [TopologicalSpace γ] {f : α → γ} (hf : AEStronglyMeasurable f μ) :
    AEStronglyMeasurable (fun z : α × β => f z.1) (μ.prod ν) :=
  hf.comp_quasiMeasurePreserving quasiMeasurePreserving_fst

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