I was trying to solve the exercise of the book "E. Cinlar – Probability and Stochastics" on page 18, No. 3.13 (c) but on the second part I lack ideas.
EX. Let $E$ be a countable set and $\mathcal{E}=2^E$. Show that every measure, $\mu$ on $(E,\mathcal{E})$ is $\Sigma$-finite, i.e. there is a sequence $(\mu_n)_n$ of measures on $(E,\mathcal{E})$ such that $\mu=\sum_{n}\mu_{n}$ and every $\mu_{n}$ is finite.
Best Answer
As stated in the example that this exercise refers to, any measure $\mu$ on $(E,\mathcal E)$ can be represented as $$ \mu=\sum_{x\in E}m(x)\delta_x,$$ where $\delta_x$ is the Dirac-measure in $x$ and $m(x):=\mu(\{x\})\in[0,\infty]$ for all $x\in E$. You can write $$ \mu=\sum_{x\in E \,: \,m(x)<\infty}m(x)\delta_x+ \sum_{x\in E \,: \,m(x)=\infty}\left(\sum_{k=1}^\infty \delta_x\right),$$ where there are countably many summands in total and each of them is a finite measure.