Helly's selection theorem


In mathematics, Helly's selection theorem states that a sequence of functions that is locally of bounded total variation and uniformly bounded at a point has a convergent subsequence. In other words, it is a compactness theorem for the space BVloc. It is named for the Austrian mathematician Eduard Helly.
The theorem has applications throughout mathematical analysis. In probability theory, the result implies compactness of a tight family of measures.

Statement of the theorem

Let U be an open subset of the real line and let fn : U → R, n ∈ N, be a sequence of functions. Suppose that
Then there exists a subsequence fnk, k ∈ N, of fn and a function f : U → R, locally of bounded variation, such that
There are many generalizations and refinements of Helly's theorem. The following theorem, for BV functions taking values in Banach spaces, is due to Barbu and Precupanu:
Let X be a reflexive, separable Hilbert space and let E be a closed, convex subset of X. Let Δ : X → . Then there exists a subsequence znk and functions δ, z ∈ BV such that