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Box-additive functions defined by measures #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

In this file we prove a few simple facts about rectangular boxes, partitions, and measures:

For the last statement, we both prove it as a proposition and define a bundled box_integral.box_additive function.

Tags #

rectangular box, measure

If μ is a locally finite measure on ℝⁿ, then λ J, (μ J).to_real is a box-additive function.

theorem {n : } (i : fin (n + 1)) (I : (fin (n + 1))) : (λ (j : fin n), (I.face i).upper j - (I.face i).lower j) * (I.upper i - I.lower i) = (λ (j : fin (n + 1)), I.upper j - I.lower j)

Box-additive map sending each box I to the continuous linear endomorphism x ↦ (volume I).to_real • x.