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 BV_loc 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.

Statement of the theorem

Let (f_n)_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 ≤ f_n ≤ b for every n ∈ N. Then the sequence (f_n)_n ∈ N admits a pointwise convergent subsequence.

Generalisation to BV_loc

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,

${\displaystyle \sup _{n\in \mathbf {N} }\left(\left\|f_{n}\right\|_{L^{1}(W)}+\left\|{\frac {\mathrm {d} f_{n}}{\mathrm {d} t}}\right\|_{L^{1}(W)}\right)<+\infty ,}$

where the derivative is taken in the sense of tempered distributions.

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 almost everywhere;
• and f_nk converges to f locally in L¹ (see locally integrable function), i.e., for all W compactly embedded in U,

${\displaystyle \lim _{k\to \infty }\int _{W}{\big |}f_{n_{k}}(x)-f(x){\big |}\,\mathrm {d} x=0;}$ ^{[1] : 132}

• and, for W compactly embedded in U,

${\displaystyle \left\|{\frac {\mathrm {d} f}{\mathrm {d} t}}\right\|_{L^{1}(W)}\leq \liminf _{k\to \infty }\left\|{\frac {\mathrm {d} f_{n_{k}}}{\mathrm {d} t}}\right\|_{L^{1}(W)}.}$ ^{[1] : 122}

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 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],

${\displaystyle \int _{[0,t)}\Delta (\mathrm {d} z_{n_{k}})\to \delta (t);}$

• 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

1. Ambrosio, Luigi; Fusco, Nicola; Pallara, Diego (2000). Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press. ISBN   9780198502456.

• 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. MR 860772