I am trying to understand the construction of the Ito integral in Section 4.3 of Kuo "Introduction to Stochastic integration", but there is one point where I am stuck: For a stochastic process $f:[a,b]\times\Omega\to\mathbb{R}$ (adapted to some filtration) which is square integrable, $\int_a^b E (|f(t,\cdot)|^2)dt<\infty$, and bounded, the author states for a $\tau\in\mathbb{R}$, $\tau\geq0$ (with $f(t)$ understood to be zero for $t<a$):
Since $f$ is assumed to be bounded, we have $$\int_a^b|f(t,\cdot)-f(t-\frac{\tau}{n},\cdot)|^2dt\to0\text{ almost surely},$$ as $n\to\infty$.
I could understand this line if $f$ would be continuous in $t$, but this is explicitly not the case. Can someone explain?
This is nothing more than the fact that translation is continuous on $L^2$, applied to the function $f_\omega: t\mapsto f(t,\omega)$ for a fixed $\omega$. As you observe, this is clear if $f_\omega$ is continuous. But any square-integrable function on the line can be approximated in the $L^2$ norm by a sequence of continuous functions of compact support.