Using exercise 9.S, we see that if $f$ is increasing and continuous on the right, we can define the Borel Stieltjes measure generated by $f$. But in the 9.T. Is it possible to define the Borel Stieltjes measure in items (c) and (d)? since the functions defined in each of the items are not increasing.
The two functions $g_3$ and $g_4$ are weakly increasing, or non decreasing. This is what the text meant with just increasing and is sufficient for the definition of the associated Borel-Stieltjes measure.
In particular, $\mu_{g_3} = \delta_0$ and $\mu_{g_4}=\mathcal{L}^1\lfloor(0,\infty)$.