Linear Combination of Real-Valued Random Variables is Real-Valued Random Variable/General Result

Theorem
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $n \in \N$.

Let $X_1, X_2, \ldots, X_n$ be real-valued random variables on $\struct {\Omega, \Sigma, \Pr}$.

Let $\alpha_1, \alpha_2, \ldots, \alpha_n$ be real numbers.

Then:


 * $\ds \sum_{i \mathop = 1}^n \alpha_i X_i$ is a real-valued random variable.