measure_compl_le_add_iff — Mathlib

English

If μ is finite and R-scalar tower structure holds between R, NNReal, and ENNReal, then r • μ is finite for any r ∈ R.

Русский

Если μ конечна и имеется структура каскадного скаляра между R, NNReal и ENNReal, то ∀ r ∈ R, r • μ конечна.

LaTeX

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

Lean4

theorem measure_compl_le_add_iff [IsFiniteMeasure μ] (hs : MeasurableSet s) (ht : MeasurableSet t) {ε : ℝ≥0∞} :
    μ sᶜ ≤ μ tᶜ + ε ↔ μ t ≤ μ s + ε :=
  ⟨fun h => compl_compl s ▸ compl_compl t ▸ measure_compl_le_add_of_le_add hs.compl ht.compl h,
    measure_compl_le_add_of_le_add ht hs⟩

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