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 ∈ 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 ≤ f_n ≤ b for every n  ∈  N. Then the sequence (f_n)_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 $A_n=\{x\in I;f_n(y)\to ̸f_n(x)\text{ as }y\to x\}$ be the set of discontinuities of $f_n$; each of these sets are countable by the above basic fact. The set $A:=\left(\bigcup _{n\in \mathbb{N}}{A}_n\right)\cup (I\cap \mathbb{Q})$ is countable, and it can be denoted as $\{a_n{\}}_{n=1}^{\mathrm{\infty }}$.

By the uniform boundedness of $\{f_n{\}}_{n=1}^{\mathrm{\infty }}$ and the Bolzano–Weierstrass theorem, there is a subsequence $\{f_n^{(1)}{\}}_{n=1}^{\mathrm{\infty }}$ such that $\{f_n^{(1)}(a_1){\}}_{n=1}^{\mathrm{\infty }}$ converges. Suppose $\{f_n^{(k)}{\}}_{n=1}^{\mathrm{\infty }}$ has been chosen such that $\{f_n^{(k)}(a_i){\}}_{n=1}^{\mathrm{\infty }}$ converges for $i=1,\dots ,k$, then by uniform boundedness and Bolzano–Weierstrass, there is a subsequence $\{f_n^{(k+1)}{\}}_{n=1}^{\mathrm{\infty }}$ of $\{f_n^{(k)}{\}}_{n=1}^{\mathrm{\infty }}$ such that $\{f_n^{(k)}(a_{k+1}){\}}_{n=1}^{\mathrm{\infty }}$ converges, thus $\{f_n^{(k+1)}(a_i){\}}_{n=1}^{\mathrm{\infty }}$ converges for $i=1,\dots ,k+1$.

Let $g_k=f_k^{(k)}$, then $\{g_k{\}}_{k=1}^{\mathrm{\infty }}$ is a subsequence of $\{f_n{\}}_{n=1}^{\mathrm{\infty }}$ that converges pointwise everywhere in $A$.

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

Let $h_k(x)=\underset{a\le x,a\in A}{\sup }{g}_k(a)$, then, h_k(a)=g_k(a) for a∈A, h_k is increasing, let $h(x)=\mathrm{lim\; sup}{\mathrm{\backslash limits}}_{k\to \mathrm{\infty }}{h}_k(x)$, then h is increasing, since supremes and limits of increasing functions are increasing, and $h(a)=\lim {\mathrm{\backslash limits}}_{k\to \mathrm{\infty }}{g}_k(a)$ 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 $g_k(q)-h(r)\le g_k(x)-h(x)\le g_k(r)-h(q)$,hence

$\mathrm{lim\; sup}{\mathrm{\backslash limits}}_{k\to \mathrm{\infty }}(g_k(x)-h(x))\le \mathrm{lim\; sup}{\mathrm{\backslash limits}}_{k\to \mathrm{\infty }}(g_k(r)-h(q))=h(r)-h(q)$

$h(q)-h(r)=\mathrm{lim\; inf}{\mathrm{\backslash limits}}_{k\to \mathrm{\infty }}(g_k(q)-h(r))\le \mathrm{lim\; inf}{\mathrm{\backslash limits}}_{k\to \mathrm{\infty }}(g_k(x)-h(x))$

Thus,

$h(q)-h(r)\le \mathrm{lim\; inf}{\mathrm{\backslash limits}}_{k\to \mathrm{\infty }}(g_k(x)-h(x))\le \mathrm{lim\; sup}{\mathrm{\backslash limits}}_{k\to \mathrm{\infty }}(g_k(x)-h(x))\le h(r)-h(q)$

Since h is continuous at x, by taking the limits $q\uparrow x,r\downarrow x$, we have $h(q),h(r)\to h(x)$, thus $\lim {\mathrm{\backslash limits}}_{k\to \mathrm{\infty }}{g}_k(x)=h(x)$

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 ∈ N that converges pointwise in I.

Generalisation to BV_loc

Let U be an open subset of the real line and let f_n : U → R, n ∈ 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 ⊆ U with compact closure W̄ ⊆ U,

$\underset{n\in 𝐍}{\sup }(\Vert f_n{\Vert }_{L^1(W)}+\Vert \frac{\mathrm{d}{f}_n}{\mathrm{d}t}{\Vert }_{L^1(W)})<+\mathrm{\infty },$

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

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

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

$\underset{k\to \mathrm{\infty }}{\lim }\underset{W}{\int }|f_{n_k}(x)-f(x)|\mathrm{d}x=0;$

• and, for W compactly embedded in U,

${\Vert \frac{\mathrm{d}f}{\mathrm{d}t}\Vert }_{L^1(W)}\le \underset{k\to \mathrm{\infty }}{\mathrm{lim\; inf}}{\Vert \frac{\mathrm{d}{f}_{n_k}}{\mathrm{d}t}\Vert }_{L^1(W)}.$

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 z_n is a uniformly bounded sequence in BV([0, T]; X) with z_n(t) ∈ E for all n ∈ N and t ∈ [0, T]. Then there exists a subsequence z_nk and functions δ, z ∈ BV([0, T]; X) such that

• for all t ∈ [0, T],

$\underset{[0,t)}{\int }\Delta (\mathrm{d}{z}_{n_k})\to \delta (t);$

• and, for all t ∈ [0, T],

$z_{n_k}(t)\rightharpoonup z(t)\in E;$

• and, for all 0 ≤ s < t ≤ T,

$\underset{[s,t)}{\int }\Delta (\mathrm{d}z)\le \delta (t)-\delta (s).$

See also

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