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_{n_k}, k ∈ N, of f_n and a function f : U → R, locally of bounded variation, such that

f_{n_k} converges to f pointwise;
and f_{n_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)}}.

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_{n_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 < t ≤ T,

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

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

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Helly's selection theorem

Statement of the theorem

Generalizations

See also

References