nullMeasurableSet_le — Mathlib

English

If AEStronglyMeasurable f holds on a trimmed measure μ.trim hm, then f is AEStronglyMeasurable on μ.

Русский

Если f AEStronglyMeasurable на μ.trim hm, то f AEStronglyMeasurable на μ.

LaTeX

$$$AEStronglyMeasurable(f, \mu.trim hm) \Rightarrow AEStronglyMeasurable(f, \mu)$$$

Lean4

theorem nullMeasurableSet_le [Preorder β] [OrderClosedTopology β] [PseudoMetrizableSpace β] {f g : α → β}
    (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) : NullMeasurableSet {a | f a ≤ g a} μ :=
  by
  apply (hf.stronglyMeasurable_mk.measurableSet_le hg.stronglyMeasurable_mk).nullMeasurableSet.congr
  filter_upwards [hf.ae_eq_mk, hg.ae_eq_mk] with x hfx hgx
  change (hf.mk f x ≤ hg.mk g x) = (f x ≤ g x)
  simp only [hfx, hgx]

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