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 BVloc 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 (fn)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 ≤ fn ≤ b for every n  ∈  N.Then the sequence (fn)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=\{x\inI:f(y)\not → f(x)asy → x\}

, i.e. the set of discontinuities, then since f is increasing, any x in A satisfies

f(x-)\leqf(x)\leqf(x+)

, where

f(x-)=\lim\limitsy\uparrowf(y)

,

f(x+)=\lim\limitsy\downarrowf(y)

, hence by discontinuity,

f(x-)<f(x+)

. Since the set of rational numbers is dense in R,

\prodx[l(f(x-),f(x+)r)\capQ]

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

+)\leq
f(x
1
+)
f(x
2
, 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

An=\{x\inI;fn(y)\not → fn(x)asy\tox\}

, i.e. the discontinuities of fn,

A=(\cupn\inAn)\cup(I\capQ)

, then A is countable, and it can be denoted as .

By the uniform boundedness of (fn)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(ai))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(ak+1))n ∈ N converges, thus (f(k+1)n)n ∈ N converges for i=1,...,k+1.

Let

(k)
g
k
, then gk is a subsequence of fn that converges pointwise in A.

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

Let

hk(x)=\supagk(a)

, then, hk(a)=gk(a) for a∈A, hk is increasing, let

h(x)=\limsup\limitsk → inftyhk(x)

, then h is increasing, since supremes and limits of increasing functions are increasing, and

h(a)=\lim\limitsk → inftygk(a)

for a∈ A by Step 2. By Step 1, h has at most countably many discontinuities.

We will show that gk converges at all continuities of h. Let x be a continuity of h, q,r∈ A, q, then

gk(q)-h(r)\leqgk(x)-h(x)\leqgk(r)-h(q)

,hence

\limsup\limitsk → inftyl(gk(x)-h(x)r)\leq\limsup\limitsk → inftyl(gk(r)-h(q)r)=h(r)-h(q)

h(q)-h(r)=\liminf\limitsk → inftyl(gk(q)-h(r)r)\leq\liminf\limitsk → inftyl(gk(x)-h(x)r)

Thus,

h(q)-h(r)\leq\liminf\limitsk → inftyl(gk(x)-h(x)r)\leq\limsup\limitsk → inftyl(gk(x)-h(x)r)\leqh(r)-h(q)

Since h is continuous at x, by taking the limits

q\uparrowx,r\downarrowx

, we have

h(q),h(r) → h(x)

, thus

\lim\limitsk → inftygk(x)=h(x)

Step 4. Choosing a subsequence of gk 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 (fn)n ∈ N that converges pointwise in I.

Generalisation to BVloc

Let U be an open subset of the real line and let fn : U → R, n ∈ N, be a sequence of functions. Suppose that(fn) 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,

\supn\left(\|fn

\|
L1(W)

+\|

dfn
dt
\|
L1(W)

\right)<+infty,

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

Then, there exists a subsequence fnk, k ∈ N, of fn and a function f : U → R, locally of bounded variation, such that

\limk\intW|

f
nk

(x)-f(x)|dx=0;

[1]

\left\|

df
dt
\right\|
L1(W)

\leq\liminfk\left\|

d
f
nk
dt
\right\|
L1(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 zn is a uniformly bounded sequence in BV([0,&nbsp;''T'']; X) with zn(t) ∈ E for all n ∈ N and t ∈ [0,&nbsp;''T'']. Then there exists a subsequence znk and functions δ, z ∈ BV([0,&nbsp;''T'']; X) such that

\int[0,\Delta(d

z
nk

)\to\delta(t);

z
nk

(t)\rightharpoonupz(t)\inE;

\int[s,\Delta(dz)\leq\delta(t)-\delta(s).

See also

References

Notes and References

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

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