div_ae_eq_one — Mathlib

English

If f is a.a.e. equal to g under μ, then g ∘ f is a.e. equal to g' ∘ f if g and g' are a.e. equal under ν and f is quasi-measure-preserving.

Русский

Если f равна a.e. g по μ, то g ∘ f равна a.e. g' ∘ f, если g ≈ a.e. g' по ν и f сохраняет меру в нужной форме.

LaTeX

$$$\text{AEMeasurable}(f, μ) \ \Rightarrow\ (\text{ae } ν)\EventuallyEq g g' \Rightarrow (\text{map } f μ) \AbsolutelyContinuous ν \Rightarrow \text{ae } μ (g \circ f) (g' \circ f).$$$

Lean4

@[to_additive]
theorem div_ae_eq_one {β} [Group β] (f g : α → β) : f / g =ᵐ[μ] 1 ↔ f =ᵐ[μ] g :=
  by
  refine ⟨fun h ↦ h.mono fun x hx ↦ ?_, fun h ↦ h.mono fun x hx ↦ ?_⟩
  · rwa [Pi.div_apply, Pi.one_apply, div_eq_one] at hx
  · rwa [Pi.div_apply, Pi.one_apply, div_eq_one]

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