Question regarding measurable random variable

Question

If $X_i, i=1,\dots, d$ are $F_T$-measurable random variables, is $\Pi_{i=1}^d X_i^k$ ($k$ is constant) also $F_T$-measurable?

Answer 1

Yes it is, since $f:(x_1,...,x_d)\mapsto\prod_{i=1}^dx^k_i$ is a continuous function (I assume your random variables have values in $\mathbb{R}$) and therefore measurable we have that $f(X_1,...,X_d)$ is a measurable random variable.