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 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

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