Let $X=\mathbb{N}$ and $\mathcal{A}$ be the $\sigma-$algebra of all subsets of $\mathbb{N}$. If $(a_n)$ is a sequence of nonnegative

Question

Let $X=\mathbb{N}$ and $\mathcal{X}$ be the $\sigma-$algebra of all subsets of $\mathbb{N}$. If $(a_n)$ is a sequence of nonnegative real numbers and if we define $\mu$ by $$ \mu(\emptyset)=0; \quad \mu(E)=\sum_{n\in E}a_n, \quad E\ne\emptyset, $$ then $\mu$ is a measure on $\mathcal{X}$. Conversely, every measure on $\mathcal{X}$ is obtained in this way for some sequence $(a_n)$ in $\overline{R}^+$.

Can someone explain to me how make the reciprocal of this exercise?

Answer 1

If you have $\mu$, then $a_n$ is determined by $\mu(\{n\})$. You can check this by countable additivity.