# Generalization Of Probability Theory Exercise

by fweth   Last Updated October 11, 2018 15:20 PM

There is exercise 13.1.6. in Klenke's Wahrscheinlichkeitstheorie:

Let $$\mu$$ be a Radon measure on $$\mathbf{R}^d$$ (I think that implies that $$\mu$$ is regular) and $$A\subseteq\mathcal{B}(\mathbf{R}^d)$$ be a $$\mu$$-null set, show that for every bounded, convex, open $$C\subseteq\mathbf{R}^d$$ with $$0\in C$$ we have $$\lim_{r\to 0}\frac{\mu(x+rC)}{r^d}=0$$ $$\lambda^d$$-a.e. on $$A$$.

I could solve it successfully, but I wondered whether we really need $$C$$ to be open and $$0\in C$$. In my proof, I think I could do without the first requirement (just $$C^\circ$$ must not be empty) but I definitely needed the second one. If it's really necessary, is there some counterexample known where $$0\notin C$$ leads to failure?

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