Independence means the joint law is the product of the marginals

Remarks

The statement is a direct translation of the definition: $\mu_{(X, Y)}(A \times B) = P(X \in A, Y \in B)$ and $(\mu_X \otimes \mu_Y)(A \times B) = \mu_X(A) \, \mu_Y(B)$. Two probability measures on $E_1 \times E_2$ that agree on all rectangles $A \times B$ are equal, because the rectangles form a π-system that generates the product σ-algebra. For real random variables, it follows that $F_{(X, Y)}(x, y) = F_X(x) \, F_Y(y)$. If $X$ and $Y$ have densities, the joint density is $f_X(x) \, f_Y(y)$. The same result holds for $n$ random variables: they are independent if and only if their joint law is $\mu_{X_1} \otimes \dots \otimes \mu_{X_n}$.