First, I give the definition from Folland,
Definition: Let $(X, \mathcal{M}, \mu)$ be a measure space. If for
each $E \in \mathcal{M}$ with $\mu(E) = \infty$, there exists $F \in
\mathcal{M}$ with $F \subset E$ and $0 < \mu(F) < \infty$, $\mu$ is
called semifinite.
Now problem:
Let X be any nonempty set, $\mathcal{M} = \mathcal{P}(X)$, and $f$ any function from $X$ to $[0, \infty]$. Then $f$ determines a measure $\mu$ on $\mathcal{M}$ by the formula $\mu(E) = \sum_{x \in E} f(x)$.
In the later paragraph, Folland states that,
The reader may verify that $\mu$ is semifinite iff $f(x) < \infty$ for every $x \in X$.
I ask that, how to verify that? Thanks.
Best Answer
If $f(x)=+\infty$ for some $x$, then $\{x\}\in\cal M$ and is of infinite measure. We cannot find a subset of finite positive measure.
If $f(x)<\infty$ for all $x\in X$, and $\mu(E)=+\infty$, let $x\in E$ such that $\mu(\{x\})>0$. Then $\mu(\{x\})$ is finite, and $\{x\}\subset E$ is measurable.