A really simple question.
If $(X, M, \mu)$ be a measurable space, a measure $v$ is absolutely continuous with respect to $u$ if
$$\forall \epsilon \;\; \exists \delta \;\; if \;\;\mu(E)<\delta\, then \;\;v(E)<\epsilon $$
What is wrong with this alternate definition:
$$\forall \epsilon \;\; \exists \delta \;\; if \;\;v(E)<\epsilon, then \;\;\mu(E)<\delta\ $$
More concretely, can anyone come up with an example of two measures which satisfy the second definition of continuity but not the first and vice versa?
