Helly's Selection Theorem

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'',
::$\sup_ \left( \left\, f_ \right\, _ + \left\, \frac \right\, _ \right) < + \infty,$
: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'',  ⊆ 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 ''L''¹ (see locally integrable function), i.e., for all ''W'' compactly embedded in ''U'',
::$\lim_ \int_ \big, f_ (x) - f(x) \big, \, \mathrm x = 0;$
* and, for ''W'' compactly embedded in ''U'',
::$\left\, \frac \right\, _ \leq \liminf_ \left\, \frac \right\, _.$

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'' →  , +∞) be positive-definite and homogeneous function">homogeneous of degree one. Suppose that ''z''_''n'' is a uniformly bounded sequence in BV( , ''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( , ''T'' ''X'') such that
* for all ''t'' ∈  , ''T''
::$\int_ \Delta (\mathrm z_) \to \delta(t);$
* and, for all ''t'' ∈  , ''T''
::$z_ (t) \rightharpoonup z(t) \in E;$
* and, for all 0 ≤ ''s'' < ''t'' ≤ ''T'',
::$\int_ \Delta(\mathrm z) \leq \delta(t) - \delta(s).$

See also

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

References

*

* {{MathSciNet, id=860772

Compactness theorems
Theorems in analysis