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[edit]

Let U be an open subset of the real line and let fn : U → R, n ∈ N, be a sequence of functions. Suppose that

where the derivative is taken in the sense of tempered distributions;
  • and (fn) is uniformly bounded at a point. That is, for some t ∈ U, { fn(t) | n ∈ N } ⊆ R is a bounded set.

Then there exists a subsequence fnk, k ∈ N, of fn and a function f : U → R, locally of bounded variation, such that

  • fnk converges to f pointwise;
  • and fnk converges to f locally in L1 (see locally integrable function), i.e., for all W compactly embedded in U,
  • and, for W compactly embedded in U,


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 → [0, +∞) be positive-definite and homogeneous of degree one. Suppose that zn is a uniformly bounded sequence in BV([0, T]; X) with zn(t) ∈ E for all n ∈ N and t ∈ [0, T]. Then there exists a subsequence znk and functions δz ∈ BV([0, T]; X) such that

  • for all t ∈ [0, T],
  • and, for all t ∈ [0, T],
  • and, for all 0 ≤ s < t ≤ T,

See also[edit]


  • Barbu, V.; Precupanu, Th. (1986). Convexity and optimization in Banach spaces. Mathematics and its Applications (East European Series). 10 (Second Romanian ed.). Dordrecht: D. Reidel Publishing Co. xviii+397. ISBN 90-277-1761-3. MR860772