Expectation is Linear/General Case

Proof
From Integral of Integrable Function is Homogeneous, we have:


 * $\alpha X$ and $\beta Y$ are $\Pr$-integrable.

From Integral of Integrable Function is Additive, we have:


 * $\alpha X + \beta Y$ is $\Pr$-integrable.

From Linear Combination of Real-Valued Random Variables is Real-Valued Random Variable, we have:


 * $\alpha X + \beta Y$ is a real-valued random variable.

Then: