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 0 j = Int(R j ) cover the set, at the cost of increasing the lengths of the sides of our rectangles very slightly (an ε2n 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).
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Throughout this chapter,μ is Lebesgue measure on R n.
If A is a subset of R n, we write −1(A) instead of −1(A, μ, measure).
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© 1993 Springer-Verlag Berlin Heidelberg
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Lang, S. (1993). Local Integration of Differential Forms. In: Real and Functional Analysis. Graduate Texts in Mathematics, vol 142. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0897-6_21
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DOI: https://doi.org/10.1007/978-1-4612-0897-6_21
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-6938-0
Online ISBN: 978-1-4612-0897-6
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