Definition 8

Tag 0C9D, permanent. Cite it as [Bourbaki, Tag 0C9D].

General Topology, chapter IX, § 6. Polish spaces, Souslin spaces, Borel sets

Let X be a Hausdorff topological space. A capacity on X is a mapping f of the set \mathscr{P}(X) of all subsets of X into the extended real line \overline{\mathbf{R}}, satisfying the following conditions:

(CA_1) If A \subset B, then f(A) \leq f(B).

(CA_2) If (A_n) is any increasing sequence of subsets of X, then

f\left(\bigcup_n A_n\right) = \sup_n f(A_n).

(CA_3) If (K_n) is any decreasing sequence of compact subsets of X, then

f\left(\bigcap_n K_n\right) = \inf_n f(K_n).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Example, § 6

Cites

Nothing.