Abstract
By an interval I in Euclidean r-space (r = 1, 2,…) is meant a rectangular parallelepiped with edges parallel to the axes. It is the Cartesian product of r 1-dimensional intervals. As in the 1 -dimensional case, the r-dimensional volume of I will be denoted by |I|. Lebesgue measure in r-space is an extension of the notion of volume to a larger class of sets. Thus Lebesgue measure has a different meaning in spaces of different dimension. However, since we shall usually regard the dimension as fixed, there is no need to indicate r explicitly in our notations.
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© 1980 Springer-Verlag New York Inc.
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Oxtoby, J.C. (1980). Lebesgue Measure in r-Space. In: Measure and Category. Graduate Texts in Mathematics, vol 2. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-9339-9_3
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DOI: https://doi.org/10.1007/978-1-4684-9339-9_3
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4684-9341-2
Online ISBN: 978-1-4684-9339-9
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