of_smul_measure — Mathlib

English

If hT is dominated FinMeasAdditive for μ and C, then μ+ν preserves the dominated property.

Русский

Если hT доминированно FinMeasAdditive для μ, то μ+ν сохраняет свойство.

LaTeX

$$of_measure_le (Measure.le_add_right le_rfl) hT hC$$

Lean4

theorem of_smul_measure {c : ℝ≥0∞} (hc_ne_top : c ≠ ∞) (hT : DominatedFinMeasAdditive (c • μ) T C) :
    DominatedFinMeasAdditive μ T (c.toReal * C) :=
  by
  have h : ∀ s, MeasurableSet s → c • μ s = ∞ → μ s = ∞ :=
    by
    intro s _ hcμs
    simp only [hc_ne_top, Algebra.id.smul_eq_mul, ENNReal.mul_eq_top, or_false, Ne, false_and] at hcμs
    exact hcμs.2
  refine ⟨hT.1.of_eq_top_imp_eq_top (μ := c • μ) h, fun s hs hμs => ?_⟩
  have hcμs : c • μ s ≠ ∞ := mt (h s hs) hμs.ne
  rw [smul_eq_mul] at hcμs
  refine (hT.2 s hs hcμs.lt_top).trans (le_of_eq ?_)
  simp only [measureReal_ennreal_smul_apply]
  ring

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