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 $A_{n}=\{x\in I;f_{n}(y)\not \rightarrow f_{n}(x){\text{ as }}y\to x\}$ be the set of discontinuities of $f_{n}$; each of these sets are countable by the above basic fact. The set $A:=\left(\textstyle \bigcup _{n\in \mathbb {N} }A_{n}\right)\cup (I\cap \mathbb {Q} )$ is countable, and it can be denoted as $\{a_{n}\}_{n=1}^{\infty }$.

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

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

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

Let $h_{k}(x)=\sup _{a\leq x,a\in A}g_{k}(a)$, then, h_k(a)=g_k(a) for a∈A, h_k is increasing, let $h(x)=\limsup \limits _{k\rightarrow \infty }h_{k}(x)$, then h is increasing, since supremes and limits of increasing functions are increasing, and $h(a)=\lim \limits _{k\rightarrow \infty }g_{k}(a)$ 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 $g_{k}(q)-h(r)\leq g_{k}(x)-h(x)\leq g_{k}(r)-h(q)$,hence

$\limsup \limits _{k\rightarrow \infty }{\bigl (}g_{k}(x)-h(x){\bigr )}\leq \limsup \limits _{k\rightarrow \infty }{\bigl (}g_{k}(r)-h(q){\bigr )}=h(r)-h(q)$

$h(q)-h(r)=\liminf \limits _{k\rightarrow \infty }{\bigl (}g_{k}(q)-h(r){\bigr )}\leq \liminf \limits _{k\rightarrow \infty }{\bigl (}g_{k}(x)-h(x){\bigr )}$

Thus,

$h(q)-h(r)\leq \liminf \limits _{k\rightarrow \infty }{\bigl (}g_{k}(x)-h(x){\bigr )}\leq \limsup \limits _{k\rightarrow \infty }{\bigl (}g_{k}(x)-h(x){\bigr )}\leq h(r)-h(q)$

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

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,

$\sup _{n\in \mathbf {N} }\left(\|f_{n}\|_{L^{1}(W)}+\|{\frac {\mathrm {d} f_{n}}{\mathrm {d} t}}\|_{L^{1}(W)}\right)<+\infty ,$

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,

$\lim _{k\to \infty }\int _{W}{\big |}f_{n_{k}}(x)-f(x){\big |}\,\mathrm {d} x=0;$

• and, for W compactly embedded in U,

$\left\|{\frac {\mathrm {d} f}{\mathrm {d} t}}\right\|_{L^{1}(W)}\leq \liminf _{k\to \infty }\left\|{\frac {\mathrm {d} f_{n_{k}}}{\mathrm {d} t}}\right\|_{L^{1}(W)}.$

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],

$\int _{[0,t)}\Delta (\mathrm {d} z_{n_{k}})\to \delta (t);$

• and, for all t ∈ [0, T],

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

• and, for all 0 ≤ s < t ≤ T,

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

See also

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

References

• Rudin, W. (1976). Principles of Mathematical Analysis. International Series in Pure and Applied Mathematics (Third ed.). New York: McGraw-Hill. 167. ISBN 978-0070542358.

• Barbu, V.; Precupanu, Th. (1986). Convexity and optimization in Banach spaces. Mathematics and its Applications (East European Series). Vol. 10 (Second Romanian ed.). Dordrecht: D. Reidel Publishing Co. xviii+397. ISBN 90-277-1761-3. MR 0860772