trim_anti_measurableSpace — Mathlib

English

Trim is antitone in the σ-algebra: if m0 ≤ m1 (coarser σ-algebra), then trimming with m1 yields ≤ trimming with m0.

Русский

Обрезка анти-σ-алгебры: если σ-алгебра m0 меньше m1, то m1.trim ≤ m0.trim.

LaTeX

$$$$ m^{\mathrm{trim}}_{m_1} \le m^{\mathrm{trim}}_{m_0} \quad \text{whenever } m_0 \le m_1. $$$$

Lean4

/-- `OuterMeasure.trim` is antitone in the σ-algebra. -/
theorem trim_anti_measurableSpace {α} (m : OuterMeasure α) {m0 m1 : MeasurableSpace α} (h : m0 ≤ m1) :
    @trim _ m1 m ≤ @trim _ m0 m := by
  simp only [le_trim_iff]
  intro s hs
  rw [trim_eq _ (h s hs)]

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