Helly's selection theorem explained

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,

\sup _{n} \left( \left\| f _{n} \right\| _{L ^{1} (W)} + \left\| (d f _{n})/(d t) \right\| _{L ^{1} (W)} \right) < + inf ty,

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

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 almost everywhere;
• and f_nk converges to f locally in L¹ (see locally integrable function), i.e., for all W compactly embedded in U,

\lim _{k} \int _{W} | f _{n _{k}} (x) - f(x) | d x = 0 ; ^[1]

• and, for W compactly embedded in U,

\left\| (d f)/(d t) \right\| _{L ^{1} (W)} \leq \liminf _{k} \left\| (d f _{n _{k}})/(d t) \right\| _{L ^{1} (W)} .

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,&nbsp;+∞) be [[positive-definite]] and homogeneous of degree one. Suppose that z_n is a uniformly bounded sequence in BV([0,&nbsp;''T'']; X) with z_n(t) ∈ E for all n ∈ N and t ∈ [0,&nbsp;''T'']. Then there exists a subsequence z_nk and functions δ, z ∈ BV([0,&nbsp;''T'']; X) such that

• for all t ∈ [0,&nbsp;''T''],

\int _{[0,} \Delta ( d z _{n _{k}} ) \to \delta(t);

• and, for all t ∈ [0,&nbsp;''T''],

z _{n _{k}} (t) \rightharpoonup z(t) \in E;

• and, for all 0 ≤ s < t ≤ T,

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

See also

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

References

• Book: Rudin , W. . Principles of Mathematical Analysis. International Series in Pure and Applied Mathematics. Third. McGraw-Hill. New York. 1976. 978-0070542358. true. 167.
• Book: Barbu , V. . Precupanu, Th.. Convexity and optimization in Banach spaces. Mathematics and its Applications (East European Series). 10. Second Romanian. D. Reidel Publishing Co.. Dordrecht. 1986. 90-277-1761-3. true. xviii+397.

Notes and References

1. Book: Ambrosio . Luigi . Fusco . Nicola . Pallara . Diego . Functions of Bounded Variation and Free Discontinuity Problems . 2000 . Oxford University Press . 9780198502456 .