In my lecture, they define Itô integral for $f:[0,T]\times \mathcal \to \mathbb R$ being progressively measurable and s.t. $$\mathbb E\int_0^T|f(t,\cdot )|^2dt<\infty .$$ However, in the book of Oksendal (SDE), he defines Itô integral for $f:[0,T]\times \Omega \to \mathbb R$ s.t. $f$ is $\mathcal B([0,T])\otimes \mathcal F$ measurable, $f(t,\cdot )$ is $\mathcal F_t$ adapted and $\mathbb E\int_0^T|f(t,\cdot )|^2dt<\infty .$
I was therefore wondering, does progressively measurability is equivalent to $f:[0,T]\times \Omega \to \mathbb R$ being $\mathcal B([0,T])\otimes \mathcal F$ measurable and $f(t,\cdot )$ is $\mathcal F_t$ adapted ?
Yes, this is almost true: Theorem IV.38 from Dellacherie and Meyer Probabilities and Potential (1978) states that an adapted measurable process has a progressively measurable modification.