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 has applications throughout mathematical analysis. In probability theory, the result implies compactness of a tight family of measures.
Statement 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_{nk}, k ∈ N, of f_{n} and a function f : U → R, locally of bounded variation, such that
- f_{nk} converges to f pointwise;
- and f_{nk} 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,
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],
- and, for all t ∈ [0, T],
- and, for all 0 ≤ s < t ≤ T,
See also
References
- Barbu, V.; Precupanu, Th. (1986). Convexity and optimization in Banach spaces. Mathematics and its Applications (East European Series). 10 (Second Romanian ed.). Dordrecht: D. Reidel Publishing Co. xviii+397. ISBN 90-277-1761-3. MR 860772