## Radon-Nikodym derivative as a limit of ratios

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 … Read more