Measure

Remarks

The series in clause 2 has non-negative terms, so it has a value in $[0, \infty]$. For $x \in X$, the Dirac measure $\delta_x$, with $\delta_x(A) = 1$ if $x \in A$ and $\delta_x(A) = 0$ otherwise, is a measure. For clause 2, the point $x$ lies in at most one of the disjoint sets $A_n$, so both sides are $1$ or both are $0$. On the power set of $\N$, the function with $\nu(\varnothing) = 0$ and $\nu(A) = 1$ for $A \ne \varnothing$ is not a measure. Clause 2 fails: $\nu(\set{1} \cup \set{2}) = 1$, while $\nu(\set{1}) + \nu(\set{2}) = 2$. A measure is monotone, $\mu(A) \le \mu(B)$ when $A \subseteq B$, and countably subadditive, $\mu(\bigcup_n A_n) \le \sum_n \mu(A_n)$. A measure with $\mu(X) = 1$ is a probability measure.