I have some function L(t) and two measure, first $\nu=\sum\delta(t-t_k), t_k=k/p$ is composition of dirac measures, second is distributed linearly with density p, I guess it has form $\mu(E)=p * (length~of~E)$, where E is interval, I'm not really familiar with that theme.
How to find Lebesgue measure $\int_{-\infty}^{\infty}L(t)d(\nu - \mu)$? Or at least can you give me textbooks about that?
Thank you.
I'd say $$\int_{-\infty}^\infty L\ d(\nu-\mu)=\int_{-\infty}^\infty L\ d\nu - \int_{-\infty}^\infty L\ d\mu=\sum_{k=-\infty}^\infty L\left({k\over p}\right)- p \int_{-\infty}^\infty L(t)\ dt\ .$$