I am using Oksendal's book of Stochastic Differential Equations. We define the space $\mathcal{V}(S,T)$ as the set of fuctions, $f$, that verify
- $f(t,\omega)$ is $\mathcal{B} \times\mathcal{F}$ mesurable.
- $\mathbb{E}[\int_S^T|f(t,\omega)|^2 dt] < \infty$.
- $ f$ is $\mathcal{F}_t$-adapted.
To define the Itô Integral he approximates $f$ by simple functions. In an intermediate step, he wants to approximate $h$ a bounded function in $\mathcal{V}$ by a sequence of bounded and continous functions $g_n$, defined as:
\begin{equation} g_n(t,\omega) = \int_0^t \phi_n(t-s) h(s, \omega) ds, \end{equation} where $\phi_n$ is a non-negative function which takes non-zero values in [$\frac{-1}{n},0]$ and also $\int_{-\infty}^{\infty} \phi_n = 1.$
How can I prove that each $g_n(t, \omega)$ is $\mathcal{F}_t$-adapted?