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 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 (fn)n ∈ N be a sequence of increasing functions mapping the real line R into itself, 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.

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 W̄ ⊆ U,
 * $$\sup_{n \in \mathbf{N}} \left( \left\| f_{n} \right\|_{L^{1} (W)} + \left\| \frac{\mathrm{d} f_{n}}{\mathrm{d} t} \right\|_{L^{1} (W)} \right) < + \infty,$$
 * where the derivative is taken in the sense of tempered distributions.

Then, there exists a subsequence fn k, k ∈ N, of fn and a function f : U → R, locally of bounded variation, such that
 * fn k converges to f pointwise almost everywhere;
 * and fn k 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 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 zn k 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 &lt; t ≤ T,
 * $$\int_{[s, t)} \Delta(\mathrm{d} z) \leq \delta(t) - \delta(s).$$