Helly's selection theorem

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 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.

Contents 1 Statement of the theorem 2 Generalisation to BVloc 3 Further generalizations 4 See also 5 References Statement of the theorem Let (fn)n ∈ N be a sequence of increasing functions mapping the real line R into itself, 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.

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, {displaystyle sup _{nin mathbf {N} }left(left|f_{n}right|_{L^{1}(W)}+left|{frac {mathrm {d} f_{n}}{mathrm {d} t}}right|_{L^{1}(W)}right)<+infty ,} where the derivative is taken in the sense of tempered distributions; and (fn) is uniformly bounded at a point. That is, for some t ∈ U, { fn(t) | n ∈ N } ⊆ R is a bounded set. Then there exists a subsequence fnk, k ∈ N, of fn and a function f : U → R, locally of bounded variation, such that fnk converges to f pointwise; and fnk converges to f locally in L1 (see locally integrable function), i.e., for all W compactly embedded in U, {displaystyle lim _{kto infty }int _{W}{big |}f_{n_{k}}(x)-f(x){big |},mathrm {d} x=0;} and, for W compactly embedded in U, {displaystyle left|{frac {mathrm {d} f}{mathrm {d} t}}right|_{L^{1}(W)}leq liminf _{kto 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 → [0, +∞) be positive-definite and homogeneous of degree one. Suppose that zn is a uniformly bounded sequence in BV([0, T]; X) with zn(t) ∈ E for all n ∈ N and t ∈ [0, T]. Then there exists a subsequence znk 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 References 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 Categories: Compactness theoremsTheorems in analysis

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