TITLE: Progressively measurable vs adapted
QUESTION [23 upvotes]: I often see in stochastic calculus books the terms 'adapted process' and 'progressively measurable process'. I know there is a small difference between them (every progressively measurable process is adapted but the converse is not necessarily true) but I can't get it from the mathematical definitions.
The definition of progressively measurable process is the following:
A stochastic process $X$ defined on a filtered probability space $(\Omega ,{\mathcal F},{({{\mathcal F}_t})_{t \ge 0}},P)$ is progressively measurable with respect to ${({{\mathcal F}_t})_{t \ge 0}}$, if the function $X(s,\omega):[0,t]\times \Omega \rightarrow \mathbb{R}$ is $\cal{B}([0,t]) \times \cal{F}_t$ measurable for every $t\ge 0$.
The definition of adapted process:
A stochastic process $X$ on $(\Omega ,{{\mathcal F}},{({{\mathcal F}_t})_{t \ge 0}},P)$ is adapted to the filtration ${({{\mathcal F}_t})_{t \ge 0}}$ (or ${\mathcal F}_t$-adapted) if $X(t)\in {\mathcal F}_t$ for each $t \ge 0$.
Can someone explain me the difference in simple words? I think I understand the definition of the adapted process quite well but I'm probably confused of the role of the Borel sigma algebra in the definition of the progressively measurable process.

REPLY [4 votes]: Theorem T46 in Meyer's book "Probability and Potentials" says: For any $(X_t)$  a measurable real-valued process adapted to the family of $\sigma$-fields $(\mathcal{F}_t)$, there exists a modification of the process $(X_t)$ progressively measurable with respect to the same family $(\mathcal{F}_t)$.
In the above example, you can easily see this is true. Change $A$ to a modification $\bar{A}=\{(x,0),x\in[0,\frac{1}{2}]\}$.
There are important relations between them: (Continuous+adapted)$\rightarrow$(mean-square continuous+adapted)$\rightarrow$predictable$\rightarrow$optional$\rightarrow$progressive$\rightarrow$adapted.