restrict_preimage — Mathlib

English

If f is measure-preserving, then restricting μa to f^{-1}(s) and μb to s yields a measure-preserving map.

Русский

Если f сохраняет меру, то ограничение меры на предобраз множества s обратно сохраняется.

LaTeX

$$$hf: MeasurePreserving f μa μb \\\\Rightarrow \\\\forall s, MeasurableSet s \\\\Rightarrow MeasurePreserving f (μa.restrict (f^{-1} s)) (μb.restrict s)$$$

Lean4

theorem restrict_preimage {f : α → β} (hf : MeasurePreserving f μa μb) {s : Set β} (hs : MeasurableSet s) :
    MeasurePreserving f (μa.restrict (f ⁻¹' s)) (μb.restrict s) :=
  ⟨hf.measurable, by rw [← hf.map_eq, restrict_map hf.measurable hs]⟩

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