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 a real interval I into the real line R,
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.

Proof

Step 1. An increasing function ''f'' on an interval I has at most countably many points of discontinuity.

Let $A=\$, i.e. the set of discontinuities, then since ''f'' is increasing, any ''x'' in A ''satisfies'' $f(x^)\leq f(x)\leq f(x^)$, where $f(x^)=\lim\limits_f(y)$,$f(x^)=\lim\limits_f(y)$, hence by discontinuity, f(x^). Since the set of rational numbers is dense in R, \prod_ bigl( f(x^-), f(x^+)\bigr) \cap \mathrm /math> is non-empty. Thus the axiom of choice indicates that there is a mapping ''s'' from A to Q.

It is sufficient to show that ''s'' is injective, which implies that A has a non-larger cardinity than Q, which is countable. Suppose ''x₁,x₂''∈A, ''x₁''<''x₂'', then f(x_1^-), by the construction of ''s'', we have ''s(x₁)2)''. Thus ''s'' is injective.

Step 2. Inductive Construction of a subsequence converging at discontinuities and rationals.

Let $A_n = \$, i.e. the discontinuities of ''f''_''n,'' $A=(\cup_A_n)\cup(\mathrm\cap\mathrm)$, then A is countable, and it can be denoted as .

By the uniform boundedness of (''f''_''n'')_{''n'' ∈ N} and B-W theorem, there is a subsequence (''f⁽¹⁾''_''n'')_{''n'' ∈ N} such that (''f⁽¹⁾''_''n''''(a₁)'')_{''n'' ∈ N} converges. Suppose (''f^(k)''_''n'')_{''n'' ∈ N} has been chosen such that (''f^(k)''_''n''''(a_i)'')_{''n'' ∈ N} converges for ''i''=1,...,k, then by uniform boundedness, there is a subsequence (''f^(k+1)''_''n'')_{''n'' ∈ N} of (''f^(k)''_''n'')_{''n'' ∈ N}, such that (''f^(k+1)''_''n''''(a_k+1)'')_{''n'' ∈ N} converges, thus (''f^(k+1)''_''n'')_{''n'' ∈ N} converges for ''i''=1,...,k+1.

Let $g_k=f^_k$, then g_''k'' is a subsequence of ''f''_''n'' that converges pointwise in A.

Step 3. ''g_k'' converges in I except possibly in an at most countable set.

Let $h_k(x)=\sup_ g_k(a)$, then, ''h_k(a)=g_k(a)'' for ''a''∈A, ''h_k'' is increasing, let $h(x)=\limsup\limits_h_k(x)$, then h is increasing, since supremes and limits of increasing functions are increasing, and $h(a)=\lim\limits_g_k(a)$ for ''a''∈ A by Step 2. By Step 1, ''h'' has at most countably many discontinuities.

We will show that ''g_k'' converges at all continuities of ''h''. Let ''x'' be a continuity of ''h'', ''q,r''∈ A, ''qg_k(q)-h(r)\leq g_k(x)-h(x)\leq g_k(r)-h(q),hence

$\limsup\limits_\bigl(g_k(x)-h(x)\bigr)\leq \limsup\limits_\bigl(g_k(r)-h(q)\bigr)=h(r)-h(q)$

$h(q)-h(r)=\liminf\limits_\bigl(g_k(q)-h(r)\bigr)\leq \liminf\limits_\bigl(g_k(x)-h(x)\bigr)$

Thus,

$h(q)-h(r)\leq\liminf\limits_\bigl(g_k(x)-h(x)\bigr)\leq \limsup\limits_\bigl(g_k(x)-h(x)\bigr)\leq h(r)-h(q)$

Since ''h'' is continuous at ''x'', by taking the limits $q\uparrow x, r\downarrow x$, we have $h(q),h(r)\rightarrow h(x)$, thus $\lim\limits_g_k(x)=h(x)$

Step 4. Choosing a subsequence of ''g_k'' that converges pointwise in I

This can be done with a diagonal process similar to Step 2.

With the above steps we have constructed a subsequence of (''f''_''n'')_{''n'' ∈ N} that converges pointwise in I.

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'',
::$\sup_ \left( \, f_ \, _ + \, \frac \, _ \right) < + \infty,$
:where the derivative is taken in the sense of tempered distributions.

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 almost everywhere;
* 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 mathematical analysis