measure_mul_setLAverage — Mathlib

English

Detailed distributive form of laverage under μ+ν into parts relative to μ and ν.

Русский

Детализированная форма распределения laverage при сумме мер μ+ν на части, пропорциональные μ и ν.

LaTeX

$$$\\text{laverage}_{μ+ν}(f) = \\text{weighted-sum}(\\laverage_{μ}(f), \\laverage_{ν}(f))$$$

Lean4

theorem measure_mul_setLAverage (f : α → ℝ≥0∞) (h : μ s ≠ ∞) : μ s * ⨍⁻ x in s, f x ∂μ = ∫⁻ x in s, f x ∂μ :=
  by
  have := Fact.mk h.lt_top
  rw [← measure_mul_laverage, restrict_apply_univ]

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