Let $X(t,\omega)$ be a stochastic process:
$$ X: \mathbb{R}^+ \times \Omega \rightarrow \mathbb{R}, $$
where $(\Omega, \mathcal{F}_t, \mathcal{F}, \mu)$ is a stochastic basis.
Some definitions:
$X$ is said to be measurable if it is Borel measurable with respect to the product $\sigma$-algebra $\mathcal{B}(\mathbb{R}^+) \otimes \mathcal{F}$, where $\mathcal{B}(\mathbb{R}^+)$ is the Borel $\sigma$-algebra on $\mathbb{R}^+$.
$X$ is said to be adapted if, for all $t>0$, $X(t, \omega)$ is $\mathcal{F}_t$-measurable.
$X$ is said to be progressively measurable if, for all $T > 0$, the restriction of $X$ to $[0, T] \times \Omega$ is Borel measurable with respect to the product $\sigma$-algebra $\mathcal{B}([0, T]) \otimes \mathcal{F}_T$-measurable.
Question
Apparently an adapted and measurable process need not be progressively measurable. This necessarily means that $\mathcal{B}([0, T]) \otimes \mathcal{F}_t$ is not the $\sigma$-algebra $[0, T] \times \Omega$ inherits as a subset of $\mathbb{R}^+ \times \Omega$. Could somebody give a simple example of such a process, adapted and measurable but not progressively measurable? Or a reference for such?