Helly's selection theorem

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 BV_loc 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 (f_n)_{n Ã¢ÂˆÂˆ 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 Ã¢ÂˆÂˆ R such that a Ã¢Â‰Â¤ f_n Ã¢Â‰Â¤ b for every n  Ã¢ÂˆÂˆ  N. Then the sequence (f_n)_{n Ã¢ÂˆÂˆ 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. g_k converges in I except possibly in an at most countable set.

Let , then, h_k(a)=g_k(a) for a∈A, h_k 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 g_k converges at all continuities of h. Let x be a continuity of h, q,r∈ A, q<x<r, then ,hence

Thus,

Since h is continuous at x, by taking the limits , we have , thus

Step 3. Choosing a subsequence of g_k 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 (f_n)_{n&nbsp;∈&nbsp;N} that converges pointwise in I.

Generalisation to BV_loc

Let U be an open subset of the real line and let f_n&nbsp;:&nbsp;U&nbsp;→&nbsp;R, n&nbsp;∈&nbsp;N, be a sequence of functions. Suppose that (f_n) has uniformly bounded total variation on any W that is compactly embedded in U. That is, for all sets W&nbsp;⊆&nbsp;U with compact closure W̄&nbsp;⊆&nbsp;U,

:

where the derivative is taken in the sense of tempered distributions.

Then, there exists a subsequence f_{n_k}, k&nbsp;∈&nbsp;N, of f_n and a function f&nbsp;:&nbsp;U&nbsp;→&nbsp;R, locally of bounded variation, such that

• f_{n_k} converges to f pointwise almost everywhere;
• and f_{n_k} 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,

:

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&nbsp;:&nbsp;X&nbsp;→&nbsp;[0,&nbsp;+∞) be positive-definite and homogeneous of degree one. Suppose that z_n is a uniformly bounded sequence in BV([0,&nbsp;T];&nbsp;X) with z_n(t)&nbsp;∈&nbsp;E for all n&nbsp;∈&nbsp;N and t&nbsp;∈&nbsp;[0,&nbsp;T]. Then there exists a subsequence z_{n_k} and functions δ,&nbsp;z&nbsp;∈&nbsp;BV([0,&nbsp;T];&nbsp;X) such that

• for all t&nbsp;∈&nbsp;[0,&nbsp;T],

:

• and, for all t&nbsp;∈&nbsp;[0,&nbsp;T],

:

• and, for all 0&nbsp;≤&nbsp;s&nbsp;&lt;&nbsp;t&nbsp;≤&nbsp;T,

:

See also

• Bounded variation
• Fraňková-Helly selection theorem
• Total variation

References