Hello,

The question might be too naive. I am just not very confident about it. Let $\mu$ be a singular measure with respect to the Lebesgue's measure on $R$. We have $\int \psi \mu(d x)=0$ for any test function $\psi\in\mathcal{D}(R)$ (smooth functions with compact support). So the singular measures are vanishing distributions. Am I right? 

After reading comments by Andreas and Wong, the above statement is wrong. :-)

Thank you very much for any hints!


Anand


p.s. this is related to [my previous post][1].


  [1]: http://mathoverflow.net/questions/70146/distributions-and-measures