Give Radon-measures $\mu$ and $\nu$ such that $\mu\ll\nu$ holds but $\nu\ll\mu$ fails

Question

I'm supposed to find Radon-measures $\mu$ and $\nu$ such that $\mu\ll\nu$ holds but $\nu\ll\mu$ fails. This seems too easy because I can take $\mu\equiv0$ and $\nu$ to be any non-zero Radon-measure (for example Lebesgue measure or Dirac measure). Then $\mu\ll \nu$ but $\nu\ll\mu$ is not true. Am I right or have I messed up something?

Answer 1

Your example is correct. To have an example with two nonzero measures $\mu$ and $\nu$, we should assume that the space has at least two points $a,b$; then $\mu=\delta_a$ and $\nu=\delta_a+\delta_b$ works.