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 ${\displaystyle A=\{x\in I:f(y)\not \rightarrow f(x){\text{ as }}y\rightarrow x\}}$, i.e. the set of discontinuities, then since f is increasing, any x in A satisfies ${\displaystyle f(x^{-})\leq f(x)\leq f(x^{+})}$, where ${\displaystyle f(x^{-})=\lim \limits _{y\uparrow x}f(y)}$,${\displaystyle f(x^{+})=\lim \limits _{y\downarrow x}f(y)}$, hence by discontinuity, ${\displaystyle f(x^{-})<f(x^{+})}$. Since the set of rational numbers is dense in R, ${\displaystyle \prod _{x\in A}[{\bigl (}f(x^{-}),f(x^{+}){\bigr )}\cap \mathrm {Q} ]}$ 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 ${\displaystyle f(x_{1}^{-})<f(x_{1}^{+})\leq f(x_{2}^{-})<f(x_{2}^{+})}$, by the construction of s, we have s(x₁)<s(x₂). Thus s is injective.

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

Let ${\displaystyle A_{n}=\{x\in I;f_{n}(y)\not \rightarrow f_{n}(x){\text{ as }}y\to x\}}$, i.e. the discontinuities of f_n, ${\displaystyle A=(\cup _{n\in \mathrm {N} }A_{n})\cup (\mathrm {I} \cap \mathrm {Q} )}$, then A is countable, and it can be denoted as {a_n: n∈N}.

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 ${\displaystyle g_{k}=f_{k}^{(k)}}$, 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 ${\displaystyle h_{k}(x)=\sup _{a\leq x,a\in A}g_{k}(a)}$, then , h_k(a)=g_k(a) for a∈A, h_k is increasing, let ${\displaystyle h(x)=\limsup \limits _{k\rightarrow \infty }h_{k}(x)}$, then h is increasing, since supremes and limits of increasing functions are increasing, and ${\displaystyle h(a)=\lim \limits _{k\rightarrow \infty }g_{k}(a)}$ 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, q<x<r, then ${\displaystyle g_{k}(q)-h(r)\leq g_{k}(x)-h(x)\leq g_{k}(r)-h(q)}$,hence

${\displaystyle \limsup \limits _{k\rightarrow \infty }{\bigl (}g_{k}(x)-h(x){\bigr )}\leq \limsup \limits _{k\rightarrow \infty }{\bigl (}g_{k}(r)-h(q){\bigr )}=h(r)-h(q)}$

${\displaystyle h(q)-h(r)=\liminf \limits _{k\rightarrow \infty }{\bigl (}g_{k}(q)-h(r){\bigr )}\leq \liminf \limits _{k\rightarrow \infty }{\bigl (}g_{k}(x)-h(x){\bigr )}}$

Thus,

${\displaystyle h(q)-h(r)\leq \liminf \limits _{k\rightarrow \infty }{\bigl (}g_{k}(x)-h(x){\bigr )}\leq \limsup \limits _{k\rightarrow \infty }{\bigl (}g_{k}(x)-h(x){\bigr )}\leq h(r)-h(q)}$

Since h is continuous at x, by taking the limits ${\displaystyle q\uparrow x,r\downarrow x}$, we have ${\displaystyle h(q),h(r)\rightarrow h(x)}$, thus ${\displaystyle \lim \limits _{k\rightarrow \infty }g_{k}(x)=h(x)}$

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 BVloc

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,

${\displaystyle \sup _{n\in \mathbf {N} }\left(\|f_{n}\|_{L^{1}(W)}+\|{\frac {\mathrm {d} f_{n}}{\mathrm {d} t}}\|_{L^{1}(W)}\right)<+\infty ,}$

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,

${\displaystyle \lim _{k\to \infty }\int _{W}{\big |}f_{n_{k}}(x)-f(x){\big |}\,\mathrm {d} x=0;}$ ^[1]: 132

• and, for W compactly embedded in U,

${\displaystyle \left\|{\frac {\mathrm {d} f}{\mathrm {d} t}}\right\|_{L^{1}(W)}\leq \liminf _{k\to \infty }\left\|{\frac {\mathrm {d} f_{n_{k}}}{\mathrm {d} t}}\right\|_{L^{1}(W)}.}$^[1]: 122

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],

${\displaystyle \int _{[0,t)}\Delta (\mathrm {d} z_{n_{k}})\to \delta (t);}$

• and, for all t ∈ [0, T],

${\displaystyle z_{n_{k}}(t)\rightharpoonup z(t)\in E;}$

• and, for all 0 ≤ s < t ≤ T,

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

See also

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