In Evan's book "An Introduction to Stochastic Differential Equations", the space $\mathbb{L}^2(0,T)$ denotes the space of progressively measurable stochastic processes $G$ such that:
$E(\int_{0}^TG^2dt)<+\infty$
and later the author introduces a larger class of integrands, $\mathbb{M}^2(0,T)$, the space of progressively measurable stochastic processes $G$ such that:
$\int_{0}^TG^2dt<+\infty$ (a.s).
I do not understand why $\mathbb{M}^2(0,T)$ is larger than $\mathbb{L}^2(0,T)$.
If $|Y|<+\infty$ a.s. then $E(|Y|)<+\infty$ ?
$$\left(\operatorname{E}[\vert X \vert] < \infty \implies \vert X \vert \underset{\text{a.s.}}{<} \infty\right) \text{ but } \left(\vert X \vert \underset{\text{a.s.}}{<} \infty \kern.6em\not\kern -.6em \implies \operatorname{E}[\vert X \vert] < \infty\right)$$
To see the latter non-implication, let $X$ be equal to $2^n$ with probability $2^{-n}$, for $n=1,2,3,\ldots$.
$X$ is finite almost surely, but its expectation diverges.