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Mathlib.MeasureTheory.Integral.IntervalAverage

Integral average over an interval #

In this file we introduce notation ⨍ x in a..b, f x for the average ⨍ x in Ι a b, f x of f over the interval Ι a b = Set.Ioc (min a b) (max a b) w.r.t. the Lebesgue measure, then prove formulas for this average:

We also prove that ⨍ x in a..b, f x = ⨍ x in b..a, f x, see interval_average_symm.

Notation #

⨍ x in a..b, f x: average of f over the interval Ι a b w.r.t. the Lebesgue measure.

source
def «term⨍_In_.._,_».«delab_app.MeasureTheory.average» :
Lean.PrettyPrinter.Delaborator.Delab

Pretty printer defined by notation3 command.

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    def «term⨍_In_.._,_» :
    Lean.ParserDescr

    ⨍ x in a..b, f x is the average of f over the interval `Ι a w.r.t. the Lebesgue measure.

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    Instances For
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      theorem interval_average_symm {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] (f : ℝ → E) (a b : ℝ) :
      ⨍ (x : ℝ) in a..b, f x = ⨍ (x : ℝ) in b..a, f x
      source
      theorem interval_average_eq {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] (f : ℝ → E) (a b : ℝ) :
      ⨍ (x : ℝ) in a..b, f x = (b - a)⁻¹ • ∫ (x : ℝ) in a..b, f x
      source
      theorem interval_average_eq_div (f : ℝ → ℝ) (a b : ℝ) :
      ⨍ (x : ℝ) in a..b, f x = (∫ (x : ℝ) in a..b, f x) / (b - a)
      source
      theorem intervalAverage_congr_codiscreteWithin {a b : ℝ} {f₁ f₂ : ℝ → ℝ} (hf : f₁ =ᶠ[Filter.codiscreteWithin (Set.uIoc a b)] f₂) :
      ⨍ (x : ℝ) in a..b, f₁ x = ⨍ (x : ℝ) in a..b, f₂ x

      Interval averages are invariant when functions change along discrete sets.