# Helly's selection theorem

In mathematics, Helly's selection theorem states that a sequence of functions that is locally of bounded total variation and uniformly bounded at a point has a convergent subsequence. In other words, it is a compactness theorem for the space BVloc. It is named for the Austrian mathematician Eduard Helly.

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 U be an open subset of the real line and let fn : U → R, n ∈ N, be a sequence of functions. Suppose that

${\displaystyle \sup _{n\in \mathbb {N} }\left(\left\|f_{n}\right\|_{L^{1}(W)}+\left\|{\frac {\mathrm {d} f_{n}}{\mathrm {d} t}}\right\|_{L^{1}(W)}\right)<+\infty ,}$
where the derivative is taken in the sense of tempered distributions;
• and (fn) is uniformly bounded at a point. That is, for some t ∈ U, { fn(t) | n ∈ N } ⊆ R is a bounded set.

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

• fnk converges to f pointwise;
• and fnk converges to f locally in L1 (see locally integrable function), i.e., for all W compactly embedded in U,
${\displaystyle \lim _{k\to \infty }\int _{W}{\big |}f_{n_{k}}(x)-f(x){\big |}\,\mathrm {d} x=0;}$
• and, for W compactly embedded in U,
${\displaystyle \left\|{\frac {\mathrm {d} f}{\mathrm {d} t}}\right\|_{L^{1}(W)}\leq \liminf _{k\to \infty }\left\|{\frac {\mathrm {d} f_{n_{k}}}{\mathrm {d} t}}\right\|_{L^{1}(W)}.}$

## 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 δz ∈ BV([0, T]; X) such that

• for all t ∈ [0, T],
${\displaystyle \int _{[0,t)}\Delta (\mathrm {d} z_{n_{k}})\to \delta (t);}$
• and, for all t ∈ [0, T],
${\displaystyle z_{n_{k}}(t)\rightharpoonup z(t)\in E;}$
• and, for all 0 ≤ s < t ≤ T,
${\displaystyle \int _{[s,t)}\Delta (\mathrm {d} z)\leq \delta (t)-\delta (s).}$