truncation — Mathlib

English

For nonnegative f and n ≠ 0, the nth moment equals the interval integral on [0,A].

Русский

При неотрицательности f и n ≠ 0 момент равен интегралу по [0,A].

LaTeX

$$$\int x\, (\operatorname{truncation}(f,A)(x))^n \mathrm{d}\mu(x) = \int_{0}^{A} y^n \, d(\mathrm{map} \, f \mu)(y)$$$

Lean4

theorem truncation {β : Type*} [MeasurableSpace β] {ν : Measure β} {f : α → ℝ} {g : β → ℝ} (h : IdentDistrib f g μ ν)
    {A : ℝ} : IdentDistrib (truncation f A) (truncation g A) μ ν :=
  h.comp (measurable_id.indicator measurableSet_Ioc)

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