# mathlibdocumentation

measure_theory.integral.interval_average

# 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:

• interval_average_eq: ⨍ x in a..b, f x = (b - a)⁻¹ • ∫ x in a..b, f x;
• interval_average_eq_div: ⨍ x in a..b, f x = (∫ x in a..b, f x) / (b - a).

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.

theorem interval_average_symm {E : Type u_1} [normed_group E] [ E] (f : → E) (a b : ) :
⨍ (x : ) in a..b, f x = ⨍ (x : ) in b..a, f x
theorem interval_average_eq {E : Type u_1} [normed_group E] [ E] (f : → E) (a b : ) :
⨍ (x : ) in a..b, f x = (b - a)⁻¹ ∫ (x : ) in a..b, f x
theorem interval_average_eq_div (f : ) (a b : ) :
⨍ (x : ) in a..b, f x = (∫ (x : ) in a..b, f x) / (b - a)