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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?

poglhar
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1 Answers1

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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.

JakobGFF
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  • So if $\xi$ is a $\mathcal{F}_T$-measurable and $f$ is a continuous function, then $f(\xi)$ is also $\mathcal{F}_T$-measurable? – poglhar Jul 23 '23 at 03:08
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    As long as you use the Borel-sigma-algebra, yes. (If you have random variables on $\mathbb{R}$ then it is standard to use the borel sigma-algebra) – JakobGFF Jul 23 '23 at 15:11
  • Thank you. Could you please tell the references for this? – poglhar Jul 24 '23 at 02:12