Helly's selection theorem

Short description: On convergent subsequences of functions that are locally of bounded total variation

In mathematics, Helly's selection theorem (also called the Helly selection principle) states that a uniformly bounded sequence of monotone real functions admits a convergent subsequence. In other words, it is a sequential compactness theorem for the space of uniformly bounded monotone functions. It is named for the Austrian mathematician Eduard Helly. A more general version of the theorem asserts compactness of the space BV_loc of functions locally of bounded total variation that are uniformly bounded at a point.

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 (f_n)_n ∈ N be a sequence of increasing functions mapping the real line R into itself, and suppose that it is uniformly bounded: there are a,b ∈ R such that a ≤ f_n ≤ b for every n  ∈  N. Then the sequence (f_n)_n ∈ N admits a pointwise convergent subsequence.

Generalisation to BV_loc

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,

[math]\displaystyle{ \sup_{n \in \mathbf{N}} \left( \left\| f_{n} \right\|_{L^{1} (W)} + \left\| \frac{\mathrm{d} f_{n}}{\mathrm{d} t} \right\|_{L^{1} (W)} \right) \lt + \infty, }[/math]

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,

[math]\displaystyle{ \lim_{k \to \infty} \int_{W} \big| f_{n_{k}} (x) - f(x) \big| \, \mathrm{d} x = 0; }[/math]

• and, for W compactly embedded in U,

[math]\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)}. }[/math]

Further 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],

[math]\displaystyle{ \int_{[0, t)} \Delta (\mathrm{d} z_{n_{k}}) \to \delta(t); }[/math]

• and, for all t ∈ [0, T],

[math]\displaystyle{ z_{n_{k}} (t) \rightharpoonup z(t) \in E; }[/math]

• and, for all 0 ≤ s < t ≤ T,

[math]\displaystyle{ \int_{[s, t)} \Delta(\mathrm{d} z) \leq \delta(t) - \delta(s). }[/math]

See also

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

References

• Rudin, W. (1976). Principles of Mathematical Analysis. International Series in Pure and Applied Mathematics (Third ed.). New York: McGraw-Hill. p. 167. ISBN 978-0070542358. 

• 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.. p. xviii+397. ISBN 90-277-1761-3.  MR860772