Helly's selection theorem (original) (raw)

From Wikipedia, the free encyclopedia

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

[edit]

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.

The proof requires the basic facts about monotonic functions: An increasing function f on an interval I has at most countably many points of discontinuity.

Step 1. Inductive Construction of a subsequence converging at discontinuities and rationals (diagonal process).

[edit]

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\}} be the set of discontinuities of f n {\displaystyle f_{n}} ; each of these sets are countable by the above basic fact. The set A := ( ⋃ n ∈ N A n ) ∪ ( I ∩ Q ) {\displaystyle A:=\left(\textstyle \bigcup _{n\in \mathbb {N} }A_{n}\right)\cup (I\cap \mathbb {Q} )} is countable, and it can be denoted as { a n } n = 1 ∞ {\displaystyle \{a_{n}\}_{n=1}^{\infty }} .

By the uniform boundedness of { f n } n = 1 ∞ {\displaystyle \{f_{n}\}_{n=1}^{\infty }} and the Bolzano–Weierstrass theorem, there is a subsequence { f n ( 1 ) } n = 1 ∞ {\displaystyle \{f_{n}^{(1)}\}_{n=1}^{\infty }} such that { f n ( 1 ) ( a 1 ) } n = 1 ∞ {\displaystyle \{f_{n}^{(1)}(a_{1})\}_{n=1}^{\infty }} converges. Suppose { f n ( k ) } n = 1 ∞ {\displaystyle \{f_{n}^{(k)}\}_{n=1}^{\infty }} has been chosen such that { f n ( k ) ( a i ) } n = 1 ∞ {\displaystyle \{f_{n}^{(k)}(a_{i})\}_{n=1}^{\infty }} converges for i = 1 , … , k {\displaystyle i=1,\dots ,k} , then by uniform boundedness and Bolzano–Weierstrass, there is a subsequence { f n ( k + 1 ) } n = 1 ∞ {\displaystyle \{f_{n}^{(k+1)}\}_{n=1}^{\infty }} of { f n ( k ) } n = 1 ∞ {\displaystyle \{f_{n}^{(k)}\}_{n=1}^{\infty }} such that { f n ( k ) ( a k + 1 ) } n = 1 ∞ {\displaystyle \{f_{n}^{(k)}(a_{k+1})\}_{n=1}^{\infty }} converges, thus { f n ( k + 1 ) ( a i ) } n = 1 ∞ {\displaystyle \{f_{n}^{(k+1)}(a_{i})\}_{n=1}^{\infty }} converges for i = 1 , … , k + 1 {\displaystyle i=1,\dots ,k+1} .

Let g k = f k ( k ) {\displaystyle g_{k}=f_{k}^{(k)}} , then { g k } k = 1 ∞ {\displaystyle \{g_{k}\}_{k=1}^{\infty }} is a subsequence of { f n } n = 1 ∞ {\displaystyle \{f_{n}\}_{n=1}^{\infty }} that converges pointwise everywhere in A {\displaystyle A} .

Step 2. gk converges in I except possibly in an at most countable set.

[edit]

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 1. Moreover, 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 3. Choosing a subsequence of gk that converges pointwise in I

[edit]

This can be done with a diagonal process similar to Step 1.

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

Generalisation to BVloc

[edit]

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 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 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 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;} [1]: 132

• 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)}.} [1]: 122

Further generalizations

[edit]

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 n k and functions δ, z ∈ BV([0, _T_]; X) such that

• for all t ∈ [0, _T_],

∫ [ 0 , t ) Δ ( d z n k ) → δ ( t ) ; {\displaystyle \int _{[0,t)}\Delta (\mathrm {d} z_{n_{k}})\to \delta (t);} ![{\displaystyle \int _{0,t)}\Delta (\mathrm {d} z_{n_{k}})\to \delta (t);}

• and, for all t ∈ [0, _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).} ![{\displaystyle \int _{s,t)}\Delta (\mathrm {d} z)\leq \delta (t)-\delta (s).}

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

1. ^ a b 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. MR 0860772