lintegral_eq — Mathlib

English

In a suitable topology, the ess sup of the identically distributed pair is equal.

Русский

В подходящей топологии ess sup идентично распределенной пары равен друг другу.

LaTeX

$$$\text{essSup}_{μ}(f) = \text{essSup}_{ν}(g)$$$

Lean4

theorem lintegral_eq {f : α → ℝ≥0∞} {g : β → ℝ≥0∞} (h : IdentDistrib f g μ ν) : ∫⁻ x, f x ∂μ = ∫⁻ x, g x ∂ν :=
  by
  change ∫⁻ x, id (f x) ∂μ = ∫⁻ x, id (g x) ∂ν
  rw [← lintegral_map' aemeasurable_id h.aemeasurable_fst, ← lintegral_map' aemeasurable_id h.aemeasurable_snd,
    h.map_eq]

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