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Helly's selection theorem

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On convergent subsequences of functions that are locally of bounded total variation

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 ∈ I : f ( y ) ↛ f ( x )  as  y → x } {\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 f ( x − ) ≤ f ( x ) ≤ f ( x + ) {\displaystyle f(x^{-})\leq f(x)\leq f(x^{+})} , where f ( x − ) = lim y ↑ x f ( y ) {\displaystyle f(x^{-})=\lim \limits _{y\uparrow x}f(y)} , f ( x + ) = lim y ↓ x f ( y ) {\displaystyle f(x^{+})=\lim \limits _{y\downarrow x}f(y)} , hence by discontinuity, f ( x − ) < f ( x + ) {\displaystyle f(x^{-})<f(x^{+})} . Since the set of rational numbers is dense in R, ∏ x ∈ A [ ( f ( x − ) , f ( x + ) ) ∩ Q ] {\displaystyle \prod _{x\in A}} 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 − ) < f ( x 1 + ) ≤ f ( x 2 − ) < f ( x 2 + ) {\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 A n = { x ∈ I ; f n ( y ) ↛ f n ( x )  as  y → x } {\displaystyle A_{n}=\{x\in I;f_{n}(y)\not \rightarrow f_{n}(x){\text{ as }}y\to x\}} , i.e. the discontinuities of fn, A = ( ∪ n ∈ N A n ) ∪ ( I ∩ Q ) {\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 {an: n∈N}.

By the uniform boundedness of (fn)n ∈ N and B-W theorem, there is a subsequence (fn)n ∈ N such that (fn(a1))n ∈ N converges. Suppose (fn)n ∈ N has been chosen such that (fn(ai))n ∈ N converges for i=1,...,k, then by uniform boundedness, there is a subsequence (fn)n ∈ N of (fn)n ∈ N, such that (fn(ak+1))n ∈ N converges, thus (fn)n ∈ N converges for i=1,...,k+1.

Let g k = f k ( k ) {\displaystyle g_{k}=f_{k}^{(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 h k ( x ) = sup a ≤ x , a ∈ A g k ( a ) {\displaystyle h_{k}(x)=\sup _{a\leq x,a\in A}g_{k}(a)} , then , hk(a)=gk(a) for a∈A, hk is increasing, let h ( x ) = lim sup k → ∞ h k ( x ) {\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 h ( a ) = lim k → ∞ g k ( a ) {\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 gk converges at all continuities of h. Let x be a continuity of h, q,r∈ A, q<x<r, then g k ( q ) − h ( r ) ≤ g k ( x ) − h ( x ) ≤ g k ( r ) − h ( q ) {\displaystyle g_{k}(q)-h(r)\leq g_{k}(x)-h(x)\leq g_{k}(r)-h(q)} ,hence

lim sup k → ∞ ( g k ( x ) − h ( x ) ) ≤ lim sup k → ∞ ( g k ( r ) − h ( q ) ) = h ( r ) − h ( q ) {\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)}

h ( q ) − h ( r ) = lim inf k → ∞ ( g k ( q ) − h ( r ) ) ≤ lim inf k → ∞ ( g k ( x ) − h ( x ) ) {\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,

h ( q ) − h ( r ) ≤ lim inf k → ∞ ( g k ( x ) − h ( x ) ) ≤ lim sup k → ∞ ( g k ( x ) − h ( x ) ) ≤ h ( r ) − h ( q ) {\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 q ↑ x , r ↓ x {\displaystyle q\uparrow x,r\downarrow x} , we have h ( q ) , h ( r ) → h ( x ) {\displaystyle h(q),h(r)\rightarrow h(x)} , thus lim k → ∞ g k ( x ) = h ( x ) {\displaystyle \lim \limits _{k\rightarrow \infty }g_{k}(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,

sup n ∈ N ( ‖ f n ‖ L 1 ( W ) + ‖ d f n d t ‖ L 1 ( W ) ) < + ∞ , {\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 fnk, k ∈ N, of fn and a function f : U → R, locally of bounded variation, such that

lim k → ∞ ∫ W | f n k ( x ) − f ( x ) | d x = 0 ; {\displaystyle \lim _{k\to \infty }\int _{W}{\big |}f_{n_{k}}(x)-f(x){\big |}\,\mathrm {d} x=0;}
  • and, for W compactly embedded in U,
‖ d f d t ‖ L 1 ( W ) ≤ lim inf k → ∞ ‖ d f n k d t ‖ L 1 ( W ) . {\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)}.}

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 → ; X) with zn(t) ∈ E for all n ∈ N and t ∈ . Then there exists a subsequence znk and functions δ, z ∈ BV(; X) such that

  • for all t ∈ ,
∫ [ 0 , t ) Δ ( d z n k ) → δ ( t ) ; {\displaystyle \int _{[0,t)}\Delta (\mathrm {d} z_{n_{k}})\to \delta (t);}
  • and, for all t ∈ ,
z n k ( t ) ⇀ z ( t ) ∈ E ; {\displaystyle z_{n_{k}}(t)\rightharpoonup z(t)\in E;}
  • and, for all 0 ≤ s < t ≤ T,
∫ [ s , t ) Δ ( d z ) ≤ δ ( t ) − δ ( s ) . {\displaystyle \int _{[s,t)}\Delta (\mathrm {d} z)\leq \delta (t)-\delta (s).}

See also

References

  1. ^ Ambrosio, Luigi; Fusco, Nicola; Pallara, Diego (2000). Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press. doi:10.1093/oso/9780198502456.001.0001. ISBN 9780198502456.
  • Rudin, W. (1976). Principles of Mathematical Analysis. International Series in Pure and Applied Mathematics (Third ed.). New York: McGraw-Hill. 167. ISBN 978-0070542358.
  • Barbu, V.; Precupanu, Th. (1986). Convexity and optimization in Banach spaces. Mathematics and its Applications (East European Series). Vol. 10 (Second Romanian ed.). Dordrecht: D. Reidel Publishing Co. xviii+397. ISBN 90-277-1761-3. MR860772
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