Usbrth

This is in regards to constructing the Ito integral, specifically the second step of approximating bounded functions by bounded and continuous functions.

Let $(Omega, mathcalF, P)$ be a probability space and let $V = V(S,T)$ be the class of functions $f: [0,infty) times Omega to mathbbR$ such that

1. 
   $(t,omega) mapsto f(t,omega)$ is $mathcalB otimes mathcalF$ - measurable, where $mathcalB$ is the Borel $sigma$-algebra on $[0,infty)$,


2. 
   $f$ is $mathcalF_t$-adapted,


3. 
   $E[ int_S^T f(t,omega)^2 dt ] < infty$.

Oksendal's 6th ed. of "Stochastic Differential Equations," on page 27, states:
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Let $h in V$ be bounded. Then there exist bounded functions $g_nin V$ such that $g(cdot,omega)$ continuous for each $omega$ and $n$，and
$$
lim_n to infty Eleft[ int_S^T (h - g_n)^2 dtright] = 0.
$$
$Proof$:Let $|h(t,omega)|leq M$ for all $(t,omega)$.For each $n$ let $psi_n$ be a non-negative,continuous function on R such that
(i)let $psi_n=0$ for $xleq -1/n$ and $xgeq 0$
(ii)$int_-infty^inftypsi_n(x)dx=1$
Define
$$
g_n(t,omega)=int_0^t psi_n(s-t)h(s,omega)ds.
$$
Then $g_n(.,omega)$ is continuous for each $omega$ and $|g_n(t,omega)|leq M$. Since $hin mathcalV$ we can show that $g_n(t,.)$ is $mathcalF_t$-measurable for all $t$.Moreover,
$$
int_S^T (h - g_n)^2 dt = 0.~as~nrightarrow infty,~for~ each~ omega
$$

In the proof, the author constructs a convolution to make the bounded function h continuous and bounded. I'm not sure why $g_n(.,w)$ is continuous for each $omega$,and the most important,why
$$
int_S^T (h - g_n)^2 dt = 0.~as~nrightarrow infty, for~each~omega
$$