Concordant pair

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,

${\sup }_{n\in \mathbb{\mathbb{N}}}({\Vert f_n\Vert }_{L^1(W)}+{\Vert \frac{\mathrm{d}{f}_n}{\mathrm{d}t}\Vert }_{L^1(W)})<+\mathrm{\infty },$

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,

$\underset{k\to \mathrm{\infty }}{\lim }\underset{W}{\int }|f_{n_k}(x)-f(x)|\mathrm{d}x=0;$

• and, for W compactly embedded in U,

${\Vert \frac{\mathrm{d}f}{\mathrm{d}t}\Vert }_{L^1(W)}\le {\mathrm{lim\; inf}}_{k\to \mathrm{\infty }}{\Vert \frac{\mathrm{d}{f}_{n_k}}{\mathrm{d}t}\Vert }_{L^1(W)}.$

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],

$\underset{[0,t)}{\int }\mathrm{\Delta }(\mathrm{d}{z}_{n_k})\to \delta (t);$

• and, for all t ∈ [0, T],

$z_{n_k}(t)\rightharpoonup z(t)\in E;$

• and, for all 0 ≤ s < t ≤ T,

$\underset{[s,t)}{\int }\mathrm{\Delta }(\mathrm{d}z)\le \delta (t)-\delta (s).$

See also

• Bounded variation
• Fraňková-Helly selection theorem
• Total variation

References

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