integral_truncation_eq_intervalIntegral_of_nonneg

English

For integrable f and nonnegativity, the moment equals the 0..A interval integral for nth moment.

Русский

Для интегрируемой и неотрицательной функции 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 integral_truncation_eq_intervalIntegral_of_nonneg (hf : AEStronglyMeasurable f μ) {A : ℝ} (h'f : 0 ≤ f) :
    ∫ x, truncation f A x ∂μ = ∫ y in 0..A, y ∂Measure.map f μ := by
  simpa using moment_truncation_eq_intervalIntegral_of_nonneg hf one_ne_zero h'f

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