Helly's selection theorem


In mathematics, Helly's selection theorem 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.

Statement of the theorem

Let n ∈ N be a sequence of increasing functions mapping a real interval I into the real line R,
and suppose that it is uniformly bounded: there are a,b ∈ R such that a ≤ fn ≤ b for every n ∈ N.
Then the sequence n ∈ N admits a pointwise convergent subsequence.

Proof

The proof requires the basic facts about monotonic functions: An increasing function f on an interval I has at most countably many points of discontinuity.

Step 1. Inductive Construction of a subsequence converging at discontinuities and rationals (diagonal process).

Let be the set of discontinuities of ; each of these sets are countable by the above basic fact. The set is countable, and it can be denoted as.
By the uniform boundedness of and the Bolzano–Weierstrass theorem, there is a subsequence such that converges. Suppose has been chosen such that converges for, then by uniform boundedness and Bolzano–Weierstrass, there is a subsequence of such that converges, thus converges for.
Let, then is a subsequence of that converges pointwise everywhere in.

Step 2. ''gk'' converges in I except possibly in an at most countable set.

Let, then, hk=gk for a∈A, hk is increasing, let, then h is increasing, since supremes and limits of increasing functions are increasing, and for a∈ A by Step 1. Moreover, h has at most countably many discontinuities.
We will show that gk converges at all continuities of h. Let x be a continuity of h, q,r∈ A, q, then,hence
Thus,
Since h is continuous at x, by taking the limits, we have, thus

Step 3. Choosing a subsequence of ''gk'' that converges pointwise in I

This can be done with a diagonal process similar to Step 1.
With the above steps we have constructed a subsequence of n ∈ N that converges pointwise in '''I.'''

Generalisation to BVloc

Let U be an open subset of the real line and let fn : U → R, n ∈ N, be a sequence of functions. Suppose that
has uniformly bounded total variation on any W that is compactly embedded in U. That is, for all sets W ⊆ U with compact closure W̄ ⊆ U,
Then, there exists a subsequence fnk, k ∈ N, of fn and a function f : U → R, locally of bounded variation, such thatfnk converges to f pointwise almost everywhere;

Further generalizations

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