according to the Randon-Nikodym Thm when μ>>v there exist a non-negative function s.t v(E)= integral_E fdμ
so I was wondering if there exists a non-negative integrable function s.t
v(E)= integral_E fdμ but μ does not absolutely continuous w.r.t v
i start from if μ not >> v then v(E)=integral_E fdμ =1 when μ(E)=0 but i cant give out this function. need help asap
When $\nu(E) = \int_E f \, d\mu$, then $\nu \ll \mu$ immediately since $\mu(E)=0$ implies $\nu(E) = \int_E f \, d\mu = 0$. The Radon-Nikodym theorem states that the converse is also true.
By the way, $\nu \ll \mu$ is read as "$\nu$ is absolutely continuous with respect to $\mu$," not the other way around.