Helly's selection theorem

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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

$\sup_{n \in \mathbb{N}} \left( \left\| f_{n} \right\|_{L^{1} (W)} + \left\| \frac{\mathrm{d} f_{n}}{\mathrm{d} t} \right\|_{L^{1} (W)} \right) < + \infty,$
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,
$\lim_{k \to \infty} \int_{W} \big| f_{n_{k}} (x) - f(x) \big| \, \mathrm{d} x = 0;$
• and, for W compactly embedded in U,
$\left\| \frac{\mathrm{d} f}{\mathrm{d} t} \right\|_{L^{1} (W)} \leq \liminf_{k \to \infty} \left\| \frac{\mathrm{d} f_{n_{k}}}{\mathrm{d} t} \right\|_{L^{1} (W)}.$

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