In the book A probability path, a random variable is defined as a function from a probability space to $(\mathbb{R}, \mathcal{B})$ the usual borel algebra over the reals. In Proposition 3.2.6. they seem to claim that the supremum of any sequence of random variables is itself a random variable, which seems incorrect to me:
Take $X_n$ to be random variables following dirac distribution at $n$, that is $X_n = n$ is a constant function. Then there is no supremum over $\mathbb{R}$! Did they forget to require boundedness or am i missing something? Here are screenshots for reference:
At first i thought maybe they meant only for finitely many, but then they apply (i) for countably many...
