How's it going?
In an attempt to use the Radon-Nykodym theorem to bulldoze through the admission of measures by bounded variation & monotonic functions (sidestepping all that Caratheodory machinery), I have stumbled upon the question in the title.
It is clear that a strictly increasing left continuous function f may be decomposed into a continuous and a jump function, where the jump function is countably supported and hence $f=\lim_ig_c+g_{s,i}=\lim_ih_i$ where each jump function $g_{s,i}$ is finitely supported. Furthermore, it is clear that each $h_i$ sends intervals to finite unions of intervals (some of which might perhaps be degenerate); that is, it sends Borel sets to Borel sets.
I am struggling, though, to show that f sends Borel sets to Borel (or at least Lebesgue Mesurable) sets, since the support for $g_s$ might be dense (say, the rationals), and so I am not sure what control might be had over the image set f(I) in this case, where I is an interval.
Note: I had asked a much broader question about the application of the Radon-Nykodym theorem to this situation a few weeks ago; I have since made peace with my approach to the problem, and only this specific technicality remains.
Also note: Since the image $h_i(I) = J_i$ is a union of (perhaps degenerate) intervals, and the procession to $h_{i+1}(I)=J_{i+1}$ at most splits 1 interval and shifts finitely many others, it seems "intuitive" that such a process could not cause the "limit" of the $J_i$'s to leave the Borels (at least give or take a Lebesgue Measure zero set). But in attempting to remove the quotations (and the doubt!) by, say, looking at the limit of the indicator functions of these sets after giving the points of jump the natural (ascending) ordering, I am still not confident in the passage from the finite to the countably infinite.
Proposed solution: perhaps I've just been approaching this all wrong!? Such an f admits a (perhaps nonstrictly) increasing inverse; this $f^{-1}$ is Borel (or at least Lebesgue) measurable, is it not (since the pullback of a ray of reals is a ray of reals)?! In other words, $f(B) = (f^{-1})^{-1}(B)$ is Borel (or at least Lebesgue)!? Can someone confirm whether this is true or not -- if it is I will be pretty damn embarrassed.