First of all, sorry for the ambiguous title. I couldn't choose a clever title explaining my question in less than $150$ characters. If you're already familiar with the notation, please skip the next paragraph.
So, I am reading an approximation theorem in the process of defining the Ito integral as a limit of the Ito integral defined for step stochastic functions. We want to prove that every function $f \in L^2_{\mathrm{ad}}([a,b]\times\Omega)$ can be approximated by a sequence $\{g_n\}$ of step stochastic processes that converges to $f$ in $L^2(\Omega)$ where $L^2_{\mathrm{ad}}([a,b]\times\Omega)$ is the space of stochastic functions adopted to a filter $\{\mathcal{F}_t: a \leq t \leq b\}$ such that $\int_a^b \mathrm{E}{|f(t)|^2} \mathrm{d}t < \infty$ and $\Omega$ is a probability space.
The proof for the possibility of this approximation is divided into three parts: $1.$ when $\mathrm{E}(f(t)f(s))$ is continuous for $(s,t) \in [a,b]^2$, $2.$ When $f$ is bounded and $3.$ when $f$ is a general function in $f \in L^2_{\mathrm{ad}}([a,b]\times\Omega)$. At the second step, when $f$ is bounded, the author defines the sequence
$$g_n = \int_0^{n(t-a)}e^{-\tau}f(t-\frac{\tau}{n}, \omega)\mathrm{d}\tau$$
and claims that $g_n$ for any $n \in \mathbb{N}$ is a step stochastic process in $L^2_{\mathrm{ad}}([a,b]\times\Omega)$
I cannot see any of this. The choice of $g_n$ looks very random to me as well.
$1.$ Why is $g_n$ a step function? Is it constant on $[n,n+1)$ with respect to the variable time?
$2.$ Why is it adapted? and why the integral of the expectation of $|g_n|^2$ is finite?
$3.$ What does motivate this choice? It looks like the convolution of $f$ with $e^{-\tau}$ which happens to be a probability measure. What's going on exactly?