Hello,


Here is my original question: let $\mu$ be a singular measure with respect to the Lebesgue's measure on $R$. Is it true that $\int \psi \mu(d x)=0$ for any test function $\psi\in C\_c^\infty(R)$ (smooth functions with compact support)? 


Following the suggestion by André Henriques, I will put the answer in the answer box. :-)


Thanks again for everyone! :-)


Anand


p.s. this is related to [my previous post](https://mathoverflow.net/questions/70146/distributions-and-measures).



Can singular measures be viewed as vanishing distributions? (Answer No!)
After reading comments by Andreas and Wong, I understand that the above statement is wrong. Professor Gerald Edgar gives the simplest example (at least for me :-) ): delta function is a singular measure, but for any test function $\psi$ not vanishing at zero $\psi(0)\ne 0$, we have that $\int \psi \delta(d x)=\psi(0)\ne 0$.


Thanks everyone for your patience and helps! :-)


