continuous_of_measurable — Mathlib

English

A Borel-measurable group hom from a locally compact normed group to a real normed space is continuous.

Русский

Берем группу с локально компактной структурой нормы; порядок сохраняющий Бorel-измеримый гомоморфизм становится непрерывным.

LaTeX

$$A Borel-measurable group hom f : G → H with G locally compact normed and H a real normed space is continuous.$$

Lean4

/-- A Borel-measurable group hom from a locally compact normed group to a real normed space is
continuous. -/
theorem continuous_of_measurable {G H : Type*} [SeminormedAddCommGroup G] [MeasurableSpace G] [BorelSpace G]
    [LocallyCompactSpace G] [SeminormedAddCommGroup H] [MeasurableSpace H] [OpensMeasurableSpace H] [NormedSpace ℝ H]
    (f : G →+ H) (hf : Measurable f) : Continuous f :=
  let ⟨_s, hs, hbdd⟩ := f.exists_nhds_isBounded hf 0;
  f.continuous_of_isBounded_nhds_zero hs hbdd

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