Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be increasing and right-continuous and let $\mu$ be the Lebesgue measure. Show that there exist $g,h : \mathbb{R} \rightarrow \mathbb{R}$ increasing and right-continuous such that $f = g+h$, $\mu_g << \mu$, and $\mu_h \perp \mu$, where $\mu_g$ and $\mu_h$ are the Lebesgue-Stieltjes measures for $g$ and $h$, respectively.
My approach is to consider the at most countable (since $f$ is increasing) set of points $M = \{m_1, m_2, \ldots\}$ on which $f$ is discontinuous. After that, I define $j(x) = f(x) - \lim_{y \uparrow x}f(y)$ which denotes the height of a jump at $x$, so $j(m_k) > 0$ for all $k \ge 1$ and $j = 0$ otherwise.
At this point, I propose that $h(x) = \sum_{n : m_n \le x} j(m_n)$ and $g(x) = f(x) - h(x)$. Now, if $h$ were to be well-defined (in this case meaning finite at every $x \in \mathbb{R}$), then it seems I can prove that $g,h$ are increasing and right-continuous, and that $\mu_h \perp \mu$. I can in fact prove that $g$ is continuous, from which I believe I can derive that $\mu_g < < \mu$. However, I realize that $h$ need not be well-defined; for example, if $f$ has a jump discontinuity of constant height at every integer.
I am not sure how else to approach the problem, and I am beginning to think that I do not even have the necessary assumptions. My best guess is that I am missing the fact that $f$ is of bounded variation, or that I may be able to do something with the Lebesgue-Radon-Nikodym theorem (although I don't see how I can derive the functions $g,h$ from the theorem). I also can't seem to find any similar problems other than ones where $f$ is a simple or bounded piecewise function.
Any help is appreciated!