setLAverage_const — Mathlib

English

For μ finite and s subset, the double integral of laverage over s equals the integral of f over s.

Русский

Для μ конечной и подмножества s равно двойной интеграл лаverage по s и интеграл f по s.

LaTeX

$$$\\int⁻_{x \\in s} ⨍⁻ a \\in s, f a \\, dμ \\; dμ = ∫⁻_{x \\in s} f x \\, dμ$$$

Lean4

theorem setLAverage_const (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) (c : ℝ≥0∞) : ⨍⁻ _x in s, c ∂μ = c := by
  simp only [setLAverage_eq, lintegral_const, Measure.restrict_apply, MeasurableSet.univ, univ_inter, div_eq_mul_inv,
    mul_assoc, ENNReal.mul_inv_cancel hs₀ hs, mul_one]

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