Durrett has a theorem that says: if $X_1, X_2, ..., X_n$ are random variables then $X_1 + X_2 + ... + X_n$ are also random variables.
My issue is how to show that $F((x_1, x_2, ... , x_n)) = \sum_{i = 1}^{n} x_i$ is measurable?
Durrett says ${x_1 + x_2 + ... + x_n < a}$ is an open set $(-\infty, a)$ so then it's measurable and this is what I don't understand.
I appreciate your help.