restrict_smul — Mathlib

English

Restriction commutes with scalar multiplication: (c • μ)|_S = c • (μ|_S).

Русский

Ограничение commuting c • μ до S равно c • (μ|_S).

LaTeX

$$$ (c \cdot \mu) \restriction s = c \cdot (\mu \restriction s) $$$

Lean4

@[simp]
theorem restrict_smul {_m0 : MeasurableSpace α} {R : Type*} [SMul R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] (c : R)
    (μ : Measure α) (s : Set α) : (c • μ).restrict s = c • μ.restrict s := by
  simpa only [smul_one_smul] using (restrictₗ s).map_smul (c • 1) μ

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