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.
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.
, i.e. the set of discontinuities, then since f is increasing, any x in A satisfies
Since the set of rational numbers is dense in R,
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.
, i.e. the discontinuities of fn,
By the uniform boundedness of (fn)n ∈ N and B-W theorem, there is a subsequence (f(1)n)n ∈ N such that (f(1)n(a1))n ∈ N converges.
Suppose (f(k)n)n ∈ N has been chosen such that (f(k)n(ai))n ∈ N converges for i=1,...,k, then by uniform boundedness, there is a subsequence (f(k+1)n)n ∈ N of (f(k)n)n ∈ N, such that (f(k+1)n(ak+1))n ∈ N converges, thus (f(k+1)n)n ∈ N converges for i=1,...,k+1.
, then gk is a subsequence of fn that converges pointwise in A.
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 lim inf lim inf lim inf Since h is continuous at x, by taking the limits 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. 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, Then, there exists a subsequence fnk, k ∈ N, of fn and a function f : U → R, locally of bounded variation, such that 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