Measurability of the supremum of a Brownian motion After reading some text books about Brownian Motion i often encountered the following object
$$
\sup_{t \in [0, T]} B_t,
$$
where $(B_t)_{t \geq 0}$ is a Brownian Motion. 
But how do i see that this object is measurable? The problem is that the supremum is taken over an uncountable set. Do i miss something trivial or why is no text book mentioning why this object is well-defined as a random variable?

You need to use the fact that $B_t$ depends continuously on $t$. That shows that the sup is the same as the sup restricted to rational $t$.
