I am studying stochastic calculus (Ito integrals, to be precise) , and I am not sure if I got some things right. For instance, we have defined $\Lambda_B ^2 (a,b)$ as the space of progressively measurable processes $X$ such that $P \left( \int_a ^b X_s ^2 ds < \infty \right)=1$.
Now, I am not sure about the meaning of $\int_a ^b X_s ^2 ds $. Let's say $X: [a,b] \times \Omega \to \mathbb{R}$ such that $X(t,\omega)=X_t (\omega)$. I think that with $\int_a ^b X_s ^2 ds $, we consider a r.v on $\Omega$ defined as:
$$ \left( \int_a ^b X_s ^2 ds \right)(\omega) = \int_a ^b X_t^2 (\omega) ds$$
If I had a stochastic integral like $\int_a ^b X_s dB_s $, the previous definition shouldn't work because it is defined as a limit of elementary processes in $L^2$, so its value in a point $\omega$ is not well defined. So to sum up when I integrate in $dt$ I end up with a pointwise well-defined r.v. on $\Omega$, while when I integrate in $dB_t$ I end up with a r.v. that, being defined as limit in $L^2$ doesn't have a well-defined pointwise value.
Is this interpretation right? If so, do you have some suggestions to further understand this topic?