Definition:Measure (Measure Theory)

A measure on a set $$X \ $$ is a function $$m:\mathcal{P} \left({X}\right) \to \R_{\geq0}$$. There may be $$A \in \mathcal{P} \left({X}\right)$$ such that $$m(A) \ $$ is undefined, and there may be $$x \in \R \ $$ such that no $$S \in  \mathcal{P} \left({S}\right)$$ satisfies $$m(S)=x \ $$.