Is the supremum of an almost surely continuous random function random variable?

Question

Let {$X_t, t\in[0,1]$} on {$R, \mathfrak B(R) $} be random, almost surely continuous, function. How to show that $X^+=sup_{t \in[0,1]} X_t$ is random variable ?

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Perhaps here I can say that $X_t$ it will be a random variable $\forall t$ аnd prove the statement like for random variables ?

Answer 1

$X^{+}=\sup\{X_t: 0\leq t \leq 1,t\in \mathbb Q\}$ outside a null set, so $X^{+}$ is almost everywhere equal to a random variable. So $X^{+}$ is Lebesgue measurable. It need not be measurable w.r.t the Borel sigma field.