Let $(\Omega ,\mathcal F,\{\mathcal F_t\},\mathbb P)$ a probability space. In the book of Oksendal (SDE), they define the Itô integral on $\mathcal V(0,T)$ being the set of stochastic process $f:[0,\infty )\times \Omega \to \mathbb R$ being :
(1) $\mathcal B\otimes \mathcal F-$measurable (where $\mathcal B$ is the Borel $\sigma-$algebra of $[0,\infty )$).
(2) $\mathcal F_t-$adapted
(3) $\mathbb E\int_0^T f^2(s,\cdot )ds<\infty $.
In the book of Baldi (stochastic calculus), he define the Itô integral on $\mathcal M^2(0,T)$ being the set of stochastic process $f:[0,\infty )\times \Omega \to \mathbb R$ being
(a) Progressively measurable
(b) $\mathbb E\int_0^T f(s,\cdot )^2ds<\infty $.
Question : Does (1) and (2) is equivalent to (a) ? I have the impression that (a) is stronger than (1) and (2), but I may be wrong. Indeed, even if $f$ is $\mathcal B\otimes \mathcal F$ measurable and $f(t,\cdot )$ is $\mathcal F_t$ measurable, I'm not so sure why $f|_{[0,t]\times \Omega }$ would be $\mathcal B([0,t])\otimes \mathcal F_t$ measurable. Any idea ?
I denote $\mathcal B(A)$ the borel $\sigma -$algebra of the set $A$.