Helly's selection theorem

In mathematics, Helly's selection theorem states that a sequence of functions that is locally of bounded total variation and uniformly bounded at a point has a convergent subsequence. In other words, it is a compactness theorem for the space BV_loc.

It is named for the Austrian mathematician Eduard Helly.

The theorem is used in Game theory, in particular when considering Games on the unit square.

tatement of the theorem

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",:::where the derivative is taken in the sense of tempered distributions;
* and ("f"_"n") is uniformly bounded at a point. That is, for some "t" ∈ "U", { "f"_"n"("t") | "n" ∈ N } ⊆ R is a bounded set.

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;
* and "f"_"n""k" converges to "f" locally in "L"¹ (see locally integrable function), i.e., for all "W" compactly embedded in "U",::
* and, for "W" compactly embedded in "U",::$left|\; frac\{mathrm\{d\}\; f\}\{mathrm\{d\}\; t\}\; ight|\_\{L^\{1\}\; (W)\}\; leq\; liminf\_\{k\; o\; infty\}\; left|\; frac\{mathrm\{d\}\; f\_\{n\{mathrm\{d\}\; t\}\; ight|\_\{L^\{1\}\; (W)\}.$

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 Banach 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"] ,::$int\_\{\; [0,\; t)\}\; Delta\; (mathrm\{d\}\; z\_\{n\_\{k)\; o\; delta(t);$
* and, for all "t" ∈ [0, "T"] ,::$z\_\{n\_\{k\; (t)\; ightharpoonup\; z(t)\; in\; E;$
* and, for all 0 ≤ "s" < "t" ≤ "T",::$int\_\{\; [s,\; t)\}\; Delta(mathrm\{d\}\; z)\; leq\; delta(t)\; -\; delta(s)$

See also

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

References

* cite book
last = Barbu
first = V.
coauthors = Precupanu, Th.
title = Convexity and optimization in Banach spaces
series = Mathematics and its Applications (East European Series)
volume = 10
edition = Second Romanian Edition
publisher = D. Reidel Publishing Co.
location = Dordrecht
year = 1986
pages = xviii+397
isbn = 90-277-1761-3 MathSciNet|id=860772