Expectation of a product of independent random variables ($\mathbb{E}{\prod }_i{X}_i={\prod }_i\mathbb{E}{X}_i$)

For independent integrable variables, $\mathbb{E}{\prod }_i{X}_i={\prod }_i\mathbb{E}{X}_i$. Tonelli theorem first gives $\mathbb{E}{\prod }_i|X_i|={\prod }_i\mathbb{E}|X_i|<\infty$; then Fubini's theorem factors the signed joint integral.