Helly's selection theorem
On convergent subsequences of functions that are locally of bounded total variation / From Wikipedia, the free encyclopedia
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In mathematics, Helly's selection theorem (also called the Helly selection principle) states that a uniformly bounded sequence of monotone real functions admits a convergent subsequence. In other words, it is a sequential compactness theorem for the space of uniformly bounded monotone functions. It is named for the Austrian mathematician Eduard Helly. A more general version of the theorem asserts compactness of the space BVloc of functions locally of bounded total variation that are uniformly bounded at a point.
The theorem has applications throughout mathematical analysis. In probability theory, the result implies compactness of a tight family of measures.