Helly's selection theorem

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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.

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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 (fn)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 (fn)n  N admits a pointwise convergent subsequence.

Proof

Step 1. An increasing function f on an interval I has at most countably many points of discontinuity.

Let , i.e. the set of discontinuities, then since f is increasing, any x in Asatisfies, where ,, hence by discontinuity, . Since the set of rational numbers is dense in R, is non-empty. Thus the axiom of choice indicates that there is a mapping s from A to Q.

It is sufficient to show that s is injective, which implies that A has a non-larger cardinity than Q, which is countable. Suppose x1,x2A, x1<x2, then , by the construction of s, we have s(x1)<s(x2). Thus s is injective.

Step 2. Inductive Construction of a subsequence converging at discontinuities and rationals.

Let , i.e. the discontinuities of fn,, then A is countable, and it can be denoted as {an: nN}.

By the uniform boundedness of (fn)n  N and B-W theorem, there is a subsequence (f(1)n)n  N such that (f(1)n(a1))n  N converges. Suppose (f(k)n)n  N has been chosen such that (f(k)n(ai))n  N converges for i=1,...,k, then by uniform boundedness, there is a subsequence (f(k+1)n)n  N of (f(k)n)n  N, such that (f(k+1)n(ak+1))n  N converges, thus (f(k+1)n)n  N converges for i=1,...,k+1.

Let , then gk is a subsequence of fn that converges pointwise in A.

Step 3. gk converges in I except possibly in an at most countable set.

Let , then, hk(a)=gk(a) for aA, hk is increasing, let , then h is increasing, since supremes and limits of increasing functions are increasing, and for aA by Step 2. By Step 1, 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<x<r, then ,hence

Thus,

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

Step 4. Choosing a subsequence of gk that converges pointwise in I

This can be done with a diagonal process similar to Step 2.

With the above steps we have constructed a subsequence of (fn)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 (fn) has uniformly bounded total variation on any W that is compactly embedded in U. That is, for all sets W  U with compact closure   U,

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

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

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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  [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 δ, z  BV([0, T]; X) such that

See also

References

  1. 1 2 Ambrosio, Luigi; Fusco, Nicola; Pallara, Diego (2000). Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press. doi:10.1093/oso/9780198502456.001.0001. ISBN   9780198502456.