aemeasurable_of_aemeasurable_exp — Mathlib

English

If exp(f) is almost everywhere measurable, then f is a.e. measurable.

Русский

Если exp(f) измеримо почти нигде не завися от меры, то f измеримо почти везде.

LaTeX

$$$$\\text{AEMeasurable}(f) \\Rightarrow \\text{AEMeasurable}(f).$$$$

Lean4

theorem aemeasurable_of_aemeasurable_exp (hf : AEMeasurable (fun x ↦ exp (f x)) μ) : AEMeasurable f μ :=
  by
  have : f = fun x ↦ log (exp (f x)) := by ext; rw [log_exp]
  rw [this]
  exact measurable_log.comp_aemeasurable hf

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