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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