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[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*,

- 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[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*_{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[edit]

## References[edit]

- 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. MR860772