IsFiniteMeasure_comap — Mathlib

English

Inverse equivalence: IsFiniteMeasure μ map f iff IsFiniteMeasure μ, under AE measurability.

Русский

Обратная эквивалентность: IsFiniteMeasure μ map f эквивалентно IsFiniteMeasure μ при AEMeasurable.

LaTeX

$$IsFiniteMeasure (μ.map f) ↔ IsFiniteMeasure μ$$

Lean4

instance IsFiniteMeasure_comap (f : β → α) [IsFiniteMeasure μ] : IsFiniteMeasure (μ.comap f) where
  measure_univ_lt_top :=
    by
    by_cases hf : Injective f ∧ ∀ s, MeasurableSet s → NullMeasurableSet (f '' s) μ
    · rw [Measure.comap_apply₀ _ _ hf.1 hf.2 MeasurableSet.univ.nullMeasurableSet]
      exact measure_lt_top μ _
    · rw [Measure.comap, dif_neg hf]
      exact zero_lt_top

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