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 is totally bounded. So one could ask: what is the smallest radius that allows to cover the set with finitely many balls?
Formally, we start with a metric space M and a subset X. The ball measure of non-compactness is defined as
Since a ball of radius r 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:
Measures of non-compactness are most commonly used if M is a normed vector space. In this case, we have in addition:
Note that these measures of non-compactness are useless for subsets of Euclidean space Rn: by the Heine-Borel theorem, every bounded closed set is compact there, which means that γ(X) = 0 or ∞ according to whether X is bounded or not.
Measures of non-compactness are however useful in the study of infinite-dimensional Banach spaces, for example. In this context, one can prove that any ball B of radius r has α(B) = r and β(B) = 2r.