Condition for Independence from Product of Expectations/Corollary/General Result

Corollary to Condition for Independence from Product of Expectations
Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space. Let $X_1, X_2, \ldots, X_n$ be independent discrete random variables.

Then:
 * $\displaystyle E \left({\prod_{k \mathop = 1}^n {X_k}}\right) = \prod_{k \mathop = 1}^n E \left({X_k}\right)$

assuming the latter expectations exist.

Proof
An inductive proof can be used.