Let $X = \{X_t, \mathscr{F}_t; 0 \le t < \infty \}$ be a progressively measurable process, and let $T$ be a stopping time of $\mathscr{F}_t$. With $f(t,x): [0, \infty) \times \mathbb{R}^d \to \mathbb{R}$ as a bounded, $\mathscr{B}([0, \infty)) \otimes \mathscr{B}(\mathbb{R}^d)$ measurable function, show that the process $Y_t = \int_0^t f(s, X_s) \, ds; t \ge 0$ is progressively measurable with respect to $\{ \mathscr{F}_t \}$ and $Y_T$ is an $\mathscr{F}_T$ measurable random variable.
If I can show that $Y_t$ is adapted to $\mathscr{F}_t$ then the rest is fairly simple.
Since each sample path is the integral of a function, it is continuous, therefore all sample paths of $Y_t$ are continuous. Then by earlier proposition 1.13, which says that any adapted process with right continuous sample paths is progressively measurable, we conclude that $Y_t$ is progressively measurable.
Then given $Y_t$ is progressively measurable, by proposition 2.18, the stopped process $Y_{T \land t}$ is also progressively measurable.
To show that $Y_T$ is $\mathscr{F}_T$ measurable, we need to show that for any Borel set $B \in \mathscr{B}(\mathbb{R}^d)$ and any $t \ge 0$, that event $\{Y_T \in B\} \cap \{T \le t\}$ is $\mathscr{F}_t$ measurable. That event is equivalent to $\{Y_{T \land t} \in B\} \cap \{T \le t\}$, and measurability of that follows from progressive measurability of $Y_{T \land t}$.
However, I'm stuck on how to show that the given $Y_t$ is adapted to $\mathscr{F}_t$. Any suggestions? Thank you.