of_smul_measure — Mathlib

English

If c>0 and hT is FinMeasAdditive for μ, then FinMeasAdditive for c•μ with transformed bound holds.

Русский

Если c>0 и hT — FinMeasAdditive для μ, то FinMeasAdditive для c•μ с преобразованной границей выполняется.

LaTeX

$$of_smul_measure {c} (hc_ne_top) (hT) : FinMeasAdditive μ T C → FinMeasAdditive (c•μ) T (c.toReal * C)$$

Lean4

theorem of_smul_measure {c : ℝ≥0∞} (hc_ne_top : c ≠ ∞) (hT : FinMeasAdditive (c • μ) T) : FinMeasAdditive μ T :=
  by
  refine of_eq_top_imp_eq_top (fun s _ hμs => ?_) hT
  rw [Measure.smul_apply, smul_eq_mul, ENNReal.mul_eq_top] at hμs
  simp only [hc_ne_top, or_false, Ne, false_and] at hμs
  exact hμs.2

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