Given $X$ and $Y$ as random variable, how to prove that $aX + bY$ as random variable for all $a, b$ in $\mathbb{R}$?
(from Karr) $ \{X + Y
Here's what I got: Given $X$ & $Y$ as r.v defined on probability space, first prove that $\{X + Y< t\}$ is r.v. Suppose $X < r$ where $r \in \mathbb{Q}$. $\{X + Y < t \}$ iff there is a rational number $r$ in the interval $\{X < r < t - Y\}$ so that $\{X < r\}$ and $\{Y < t -r\}$. Hence, for all $r$, the union of pairwise disjoint, $\{X < r\} \in F$ and $\{Y < t-r\} \in F$, is in $F$. So, this will arrive to the above countable union, which proves that $X + Y$ is a random variable. To prove $aX$ and $bY$, we show that $aX$ is r.v. If $a > 0$, then for each $t \in \mathbb{R}, \{aX \leq t\} = \{X \leq t/a\}$ which is in $F$. If $a < 0$, $\{aX \leq t\} = \{X \geq t/a\}$ which is again in $F$. So, $aX$ and $bY$ are r.v. It can now be concluded that $aX + bY$ is random variable for $a, b \in \mathbb{R}$. Is this proper now?