Real and Functional Analysis pp 497-522 | Cite as

# Local Integration of Differential Forms

## Abstract

We recall that a set has **measure** 0 in **R** _{ n } if and only if, given e, there exists a covering of the set by a sequence of rectangles {*R* _{ j }} such that ∑ *μ*(*R* _{ j }) *<* ε. We denote by *R* _{ j } the closed rectangles, and we may always assume that the interiors *R* _{ j } ^{0} = Int(*R* _{ j }) cover the set, at the cost of increasing the lengths of the sides of our rectangles very slightly (an ε2^{ n } argument). We shall prove here some criteria for a set to have measure 0. We leave it to the reader to verify that instead of rectangles, we could have used cubes in our characterization of a set of a measure 0 (a cube being a rectangle all of whose sides have the same length).

### Keywords

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