Let $X$ be a countable set. Define a measure on $X$ as: $$\mu:\mathcal{P}(X)\rightarrow[0,\infty)$$
Such that $\mu(\emptyset)=0$ and for every disjoint sequence $E_n\subseteq X, n\geq 1$, we have: $$\mu(\cup_{n=1}^\infty E_n)=\sum_{i=1}^{\infty}\mu(E_n)$$
Find a description of the all possible measures on $X$; that is, show that there exists sets $A$, $B$, and a bijective map from the set of measures on $X$ to $B^A$.
I'm not exactly sure what I should be doing here. Any help is appreciated.
If $\mu:P(X)\to [0,\infty]$ that means that $$m(\{x\})=a_x\in[0,\infty]$$ for all $x\in X$ since $\{x\}\in P(X)$. But since every set $A\subset X$ can be described as $$A=\bigcup_{x\in A}\{x\}$$
we see that since the right side of that equality is countable sequence of disjoint sets we have $$\mu(A)=\mu(\bigcup_{x\in A}\{x\})=\sum\limits_{x\in A}\mu(x)=\sum\limits_{x\in A}a_x$$
or
$$\mu(A)=\sum\limits_{x\in X} a_x 1_{A}(x)$$
Thus choosing a sequence $\{a_x\}_{x\in X}$ of positive extended real numbers uniquely determines $\mu$.