Question

Hello,

This problem bothers me for some time. Suppose that

1. $\mu$ is a non-negative Radon measure (or positive linear functional of the space of continuous functions with compact support);
2. $\psi$ is a continuous function, vanishing at infinity and integrable, i.e., $\psi\in C^0_0(R)\cap L^1(R)$;
3. $\sup_{x \in R}|(\psi*\mu)(x)|<+\infty$.

Then, we would like to prove that the function $x\mapsto (\psi*\mu)(x)$ is continuous.

Thank you very much for your help and any hints!

Anand

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Version 2. If we add an additional property,

1. $\sup_{x\in R} |(G*\mu)(x)|<+\infty$, where $G(x)=\frac{1}{\sqrt{2\pi}}\exp(-x^2/2)$.

Then, is it possible to prove that the function $x\mapsto (\psi*\mu)(x)$ is continuous?

Thanks

Anand