What is the space $L^\infty (\Omega \times [0,T])$ ? Is it $\mathbb E[\sup_{t\in [0,T]}|X_t|]<\infty $?

Question

I know for example that $L^p(\Omega \times [0,T])$ is the set of stochastic process s.t. $$\mathbb E\left[\int_0^T|X_t|^p\right]<\infty .$$ But what is $L^\infty (\Omega \times [0,T])$ ? Would it be the set of stochastic process s.t. $$\mathbb E\left[\sup_{t\in [0,T]}|X_t|\right]<\infty $$ or s.t. $$\mathbb P\left\{\sup_{(\omega ,t)\in \Omega \times [0,T]}|X_t(\omega )|<\infty \right\}=1\ \ ?$$

Answer 1

Usually, $ L^\infty(X)$, where $(X,\mathcal A,\mu)$ is a measure space, is the set of all the (equivalence class of) functions $f\colon X\to R$ such that there exists an $M$ satisfying $\left\lvert f(x)\right\rvert\leqslant M$ for $\mu$-almost every $x\in X$.

Here, $X=\Omega\times [0,T]$, $\mathcal A$ is the product $\sigma$-algebra of the $\sigma$-algebra on $\Omega$ with the Borel $\sigma$-algebra on $[0,T]$ and $\mu$ is the product measure of $\mathbb P$ with the Lebesgue measure on $[0,T]$.