term⨍⁻_,_∂_ — Mathlib

English

If f is LocallyIntegrableOn a set with atTop, then integrability passes to the set via BigO.

Русский

Если f локально интегрируемо на множестве с atTop, интегрируемость передается через BigO.

LaTeX

$$$\\mathrm{LocallyIntegrableOn}\\ f\\ (Ici a)\\Rightarrow \\mathrm{IntegrableOn}\\ f\\ (Ici a)$$$

Lean4

/-- Average value of an `ℝ≥0∞`-valued function `f` w.r.t. a measure `μ`.

It is equal to `(μ univ)⁻¹ * ∫⁻ x, f x ∂μ`, so it takes value zero if `μ` is an infinite measure. If
`μ` is a probability measure, then the average of any function is equal to its integral.

For the average on a set, use `⨍⁻ x in s, f x ∂μ`, defined as `⨍⁻ x, f x ∂(μ.restrict s)`. For the
average w.r.t. the volume, one can omit `∂volume`. -/
@[term_parser 1000]
public meta def «term⨍⁻_,_∂_» : Lean.ParserDescr✝ :=
  ParserDescr.node✝ `MeasureTheory.«term⨍⁻_,_∂_» 1022
    (ParserDescr.binary✝ `andthen
      (ParserDescr.binary✝ `andthen
        (ParserDescr.binary✝ `andthen
          (ParserDescr.binary✝ `andthen
            (ParserDescr.binary✝ `andthen (ParserDescr.symbol✝ "⨍⁻") Batteries.ExtendedBinder.extBinders)
            (ParserDescr.symbol✝ ", "))
          (ParserDescr.cat✝ `term 60))
        (ParserDescr.symbol✝ " ∂"))
      (ParserDescr.cat✝ `term 70))

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