This question is related to Radon-Nikodym derivatives as limits of ratios.

Let $F$, $G$ be sigma-finite measures (or at least probability measures) on $\mathbb{R}$ such that $F \ll G$.

The theorem quoted in the link tells that the Radon-Nikodym derivative checks $$\frac{\mathrm{d}F}{\mathrm{d}G} (x) = \lim_{h\to 0^+} \frac{F(x-h, x+h)}{G(x-h, x+h)}$$ for $G$-almost every $x$.

Do we have a similar equality with one-sided balls? In other words, is the following equality $$\frac{\mathrm{d}F}{\mathrm{d}G} (x) = \lim_{h\to 0^+} \frac{F [x, x+h)}{G [x, x+h)}$$ true for $G$-almost every $x$?

Thank you very much.

Gisnotthe Lebesgue measure. Is this notion defined somewhere for more general measures? $\endgroup$Gand nicely shrinking sets, provided that "nicely shrinking" is defined with respect to the measureG, not the Lebesgue measure. Unfortunately, finding a reference to this fact is hard. The only one I have found is in this online supplementary material of the book A First Course in Sobolev Spaces. Does anybody know another reference? $\endgroup$G[x,x+h) > CG[x-h,x+h]. Ifxis a jump point ofG, it is OK. For (Lebesgue-)almost all continuity points,G'(x) exists and is the limit ofG[x,x+h) /h andG[x-h,x+h] /(2h). But does the ratioG[x,x+h) /G[x-h,x+h] converges to 1/2? It would be OK ifG'(x) were not 0 on all that points. But unfortunately that is not sure asGmight be singular... $\endgroup$