support — Mathlib

English

Definition: The support of μ is the set of x such that every neighborhood of x has positive measure.

Русский

Опора меры μ — это множества x such что каждая окрестность x имеет положительную меру.

LaTeX

$$$\\\\mathrm{supp}(\\\\mu) = \\{ x : X \\\\mid \\\\forall U \\\\text{open}, x \\in U \\Rightarrow \\\\mu(U) > 0 \\\\\\}$$$

Lean4

/-- A point `x` is in the support of `μ` if any open neighborhood of `x` has positive measure.
We provide the definition in terms of the filter-theoretic equivalent
`∃ᶠ u in (𝓝 x).smallSets, 0 < μ u`. -/
protected def support (μ : Measure X) : Set X :=
  {x : X | ∃ᶠ u in (𝓝 x).smallSets, 0 < μ u}

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