Integral average over an interval #

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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.

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theorem interval_average_symm {E : Type u_1} [normed_add_comm_group E] [normed_space ℝ E] [complete_space E] (f : ℝ → E) (a b : ℝ) :

⨍ (x : ℝ) in a..b, f x = ⨍ (x : ℝ) in b..a, f x

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theorem interval_average_eq {E : Type u_1} [normed_add_comm_group E] [normed_space ℝ E] [complete_space E] (f : ℝ → E) (a b : ℝ) :

⨍ (x : ℝ) in a..b, f x = (b - a)⁻¹ • ∫ (x : ℝ) in a..b, f x

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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)