Linear Combination of Real-Valued Random Variables is Real-Valued Random Variable

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

Let $X$ and $Y$ be real-valued random variables on $\struct {\Omega, \Sigma, \Pr}$.

Let $\alpha$ and $\beta$ be real numbers.

Then:


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

Proof
Since $X$ and $Y$ are real-valued random variables, we have:


 * $X$ and $Y$ are $\Sigma$-measurable functions.

From Pointwise Scalar Multiple of Measurable Function is Measurable, we have:


 * $\alpha X$ and $\beta Y$ are $\Sigma$-measurable.

From Pointwise Sum of Measurable Functions is Measurable, we have:


 * $\alpha X + \beta Y$ is $\Sigma$-measurable.

So:


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