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 BV_loc. 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 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,

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

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

• and (f_n) is uniformly bounded at a point. That is, for some t ∈ U, { f_n(t) | n ∈ N } ⊆ R is a bounded set.

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;
• 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,

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

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

• Barbu, V.; Precupanu, Th. (1986). Convexity and optimization in Banach spaces. Mathematics and its Applications (East European Series) 10 (Second Romanian ed.). Dordrecht: D. Reidel Publishing Co. xviii+397. ISBN 90-277-1761-3. MR 860772