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.

**Answer**

**Attribution***Source : Link , Question Author : PhilippeC , Answer Author : Community*