Factorized Joint Probability Mass Function Implies Independence

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $\mathbb{N}$ be the set of natural numbers with $\mathbb{N}_0=\mathbb{N}\cup\{0\}$, and let $r\in\mathbb{N}$. Let $X_1,\dots,X_r$ be random variables such that $X_i(\omega)\in\mathbb{N}_0$ for every $\omega\in\Omega$ and every $i$. Suppose that for each $i$ there is a function $g_i:\mathbb{N}_0\to[0,1]$ with $\sum_{c=0}^{\infty}g_i(c)=1$ such that for all $(c_1,\dots,c_r)\in\mathbb{N}_0^{r}$,

$$P\Bigl(\bigcap_{i=1}^{r}\{X_i=c_i\}\Bigr)=\prod_{i=1}^{r}g_i(c_i),$$

with the finite product notation. Then:

1. for every $i$ and every Borel set $B$,

$$P(X_i\in B)=\sum_{c\in B\cap\mathbb{N}_0}g_i(c),$$

the sum over the countable index set $B\cap\mathbb{N}_0$ being the supremum of its finite partial sums (all terms are nonnegative); in particular $P(X_i=c)=g_i(c)$;

2. the random variables $X_1,\dots,X_r$ are independent.