measure_preimage_of_map_eq_self — Mathlib

English

For a measurable f, mapₗ f μ = 0 iff μ = 0.

Русский

Для измеримой f: mapₗ f μ = 0 эквивалентно μ = 0.

LaTeX

$$$ map_\ell f \mu = 0 \iff \mu = 0 $ for measurable f$$

Lean4

/-- If `map f μ = μ`, then the measure of the preimage of any null measurable set `s`
is equal to the measure of `s`.
Note that this lemma does not assume (a.e.) measurability of `f`. -/
theorem measure_preimage_of_map_eq_self {f : α → α} (hf : map f μ = μ) {s : Set α} (hs : NullMeasurableSet s μ) :
    μ (f ⁻¹' s) = μ s := by
  if hfm : AEMeasurable f μ then
    rw [← map_apply₀ hfm, hf]
    rwa [hf]
  else
    rw [map_of_not_aemeasurable hfm] at hf
    simp [← hf]

Похожие утверждения