image_inclusion' — Mathlib

English

If s ⊆ t and hs, hu are measurable, then the image inclusion is measurable.

Русский

Если s ⊆ t и hs, hu измеримы, то образующее включение измеримо.

LaTeX

$$$\\\\operatorname{image\\\\_inclusion}'(h, hs, hu)$$$

Lean4

theorem image_inclusion' {s t : Set α} (h : s ⊆ t) {u : Set s} (hs : MeasurableSet (Subtype.val ⁻¹' s : Set t))
    (hu : MeasurableSet u) : MeasurableSet (inclusion h '' u) :=
  by
  rcases hu with ⟨u, hu, rfl⟩
  convert (measurable_subtype_coe hu).inter hs
  ext ⟨x, hx⟩
  simpa [@and_comm _ (_ = x)] using and_comm

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