In the book "Stochastic Differential Equations" by Oksendal, at the page $27$, in the last few lines he has written
Define $g_{n}(t,\omega) := \int_{0}^{t}\psi_{n}(s-t)h(s,\omega)ds$. Then, $g_{n}(.,\omega)$ is continuous for each $\omega $ & $|g_{n}(t,\omega)| \leq M$ . Since $h \in \nu$ , we can show that: $g_{n}(t,.)$ is $\mathcal F_{t}$-measurable $\forall t$.
-where: $h \in \nu$ such that: $|h(t,\omega)| \leq M ; \forall (t,\omega)$ & for each $n$, let $\psi_{n}$ be a non-negative, continuous function on $\mathbb R$ such that:
i) $\psi_{n}(x) = 0 $ for $x \leq -\frac{1}{n}$ & $x \geq 0$
ii) $\int_{\mathbb R} \psi_{n}(x) dx = 1$.
Can someone please explain to me: How $g_{n}(t,.)$ is $\mathcal F_{t}$-measurable $\forall t$??
P.S. :- As referred by the author, I have checked Karatzas & Shreeve. The setup there is bit more complicated & I couldn't find out from there.
Any detailed explanation is welcome!! Thank You!!