Infunctional analysis, twomeasures of non-compactness are commonly used; these associate numbers to sets in such a way thatcompact sets all get the measure 0, and other sets get measures that are bigger according to "how far" they are removed from compactness.

The underlying idea is the following: a bounded set can be covered by a single ball of some radius. Sometimes several balls of a smaller radius can also cover the set. A compact set in fact can be covered by finitely many balls of arbitrary small radius, because it istotally bounded. So one could ask: what is the smallest radius that allows to cover the set with finitely many balls?

Formally, we start with ametric spaceM and a subsetX. Theball measure of non-compactness is defined as

α(X) =inf {r > 0 : there exist finitely many balls of radiusr which coverX}

and theKuratowski measure of non-compactness is defined as

β(X) = inf {d > 0 : there exist finitely many sets of diameter at mostd which coverX}

Since a ball of radiusr has diameter at most 2r, we have α(X) ≤ β(X) ≤ 2α(X).

The two measures α and β share many properties, and we will use γ in the sequel to denote either one of them. Here is a collection of facts:

• X is bounded if and only if γ(X) < ∞.
• γ(X) = γ(X^cl), whereX^cl denotes theclosure ofX.
• IfX is compact, then γ(X) = 0. Conversely, if γ(X) = 0 andX iscomplete, thenX is compact.
• γ(X ∪Y) = max(γ(X), γ(Y)) for any two subsetsX andY.
• γ is continuous with respect to theHausdorff distance of sets.

Measures of non-compactness are most commonly used ifM is anormed vector space. In this case, we have in addition:

• γ(aX) = |a| γ(X) for anyscalara
• γ(X +Y) ≤ γ(X) + γ(Y)
• γ(conv(X)) = γ(X), where conv(X) denotes theconvex hull ofX

Note that these measures of non-compactness are useless for subsets ofEuclidean spaceRⁿ: by theHeine–Borel theorem, every bounded closed set is compact there, which means that γ(X) = 0 or ∞ according to whetherX is bounded or not.

Measures of non-compactness are however useful in the study of infinite-dimensionalBanach spaces, for example. In this context, one can prove that any ballB of radiusr has α(B) =r and β(B) = 2r.

• Kuratowski's intersection theorem

1. Józef Banaś,Kazimierz Goebel:Measures of noncompactness in Banach spaces, Institute of Mathematics, Polish Academy of Sciences, Warszawa 1979
2. Kazimierz Kuratowski:Topologie Vol I, PWN. Warszawa 1958
3. R.R. Akhmerov, M.I. Kamenskii, A.S. Potapova, A.E. Rodkina and B.N. Sadovskii,Measure of Noncompactness and Condensing Operators, Birkhäuser, Basel 1992

• Functional analysis