essInf_apply — Mathlib

English

If ν is absolutely continuous with respect to μ, then essSup f ν ≤ essSup f μ.

Русский

Если ν абсолютно непрерывна по отношению к μ, то essSup fν ≤ essSup fμ.

LaTeX

$$$essSup f ν ≤ essSup f μ$$$

Lean4

theorem essInf_apply {_ : MeasurableSpace α} {γ} [CompleteLattice γ] (f : α → β) (μ : Measure α) (g : β ≃o γ) :
    g (essInf f μ) = essInf (fun x => g (f x)) μ :=
  @OrderIso.essSup_apply α βᵒᵈ _ _ γᵒᵈ _ _ _ g.dual

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