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Measure zero
Let μ be a measure on a sigma algebra Σ of subsets of a set X. An element A in Σ is said to have measure zero if μ(A)=0.
Any set of measure zero is a null set. The opposite is not true, because a null set is not required to be measurable, that is, to be an element in Σ. However, any null set is a subset of a set of measure zero.
Examples
Consider the Lebesgue measure μ over the real numbers.
- Any countable set of real numbers has measure zero. In particular, the set of all rational numbers and the set of discontinuities of a monotonic function have measure zero.
- An uncountable set of real numbers which has measure zero is the Cantor set.
See also
10-26-2009 08:16:03
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The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details