measure_compl_le_add_of_le_add — Mathlib

English

If μ is finite, then for any r in NNReal, r • μ is finite.

Русский

Если μ конечна, то для любого r в NNReal, r • μ конечна.

LaTeX

$$IsFiniteMeasure( instHSMul.hSMul r μ )$$

Lean4

theorem measure_compl_le_add_of_le_add [IsFiniteMeasure μ] (hs : MeasurableSet s) (ht : MeasurableSet t) {ε : ℝ≥0∞}
    (h : μ s ≤ μ t + ε) : μ tᶜ ≤ μ sᶜ + ε :=
  by
  rw [measure_compl ht (measure_ne_top μ _), measure_compl hs (measure_ne_top μ _), tsub_le_iff_right]
  calc
    μ univ = μ univ - μ s + μ s := (tsub_add_cancel_of_le <| measure_mono s.subset_univ).symm
    _ ≤ μ univ - μ s + (μ t + ε) := (add_le_add_left h _)
    _ = _ := by rw [add_right_comm, add_assoc]

Похожие утверждения