memLp_top_of_hasCompactSupport — Mathlib

English

For a star-compatible function space, eLpNorm of the star-applied function equals eLpNorm of the function.

Русский

Для пространства AEEqFun норма eLp остаётся неизменной под действием операции star.

LaTeX

$$$$eLpNorm_{p}(star\\ f) = eLpNorm_{p}(f).$$$$

Lean4

/-- A continuous function with compact support belongs to `L^∞`.
See `Continuous.memLp_of_hasCompactSupport` for a version for `L^p`. -/
theorem _root_.Continuous.memLp_top_of_hasCompactSupport {X : Type*} [TopologicalSpace X] [MeasurableSpace X]
    [OpensMeasurableSpace X] {f : X → E} (hf : Continuous f) (h'f : HasCompactSupport f) (μ : Measure X) :
    MemLp f ⊤ μ := by
  borelize E
  rcases hf.bounded_above_of_compact_support h'f with ⟨C, hC⟩
  apply memLp_top_of_bound ?_ C (Filter.Eventually.of_forall hC)
  exact (hf.stronglyMeasurable_of_hasCompactSupport h'f).aestronglyMeasurable

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