The underlying idea is the following: a bounded set can be covered by a single ball of some radius.
A compact set in fact can be covered by finitely many balls of arbitrary small radius, because it is totally bounded.
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.