| 释义 |
Vitali set An example of a set which is not measurable, named after Giuseppe Vitali. The equivalence relation ~ is defined on the interval [0,1] by x ~ y if x–y is rational. By the axiom of choice, a set, the Vitali set, exists comprising one element from each equivalence class. It can then be shown that if the Vitali set has measure zero, then [0,1] has measure zero, and if the Vitali set has positive measure, then [–1,2] has infinite measure. Both are contradictions, and so the Vitali set is non-measurable.
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