## A small variation on the definition of $\epsilon - \delta$ absolutely continuous measure? [closed]

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?

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"Close as too localized" – Qfwfq Nov 18 2011 at 14:37
Your question would be more appropriate at math.stackexchange.com or one of the other sites listed in the FAQ. – S. Carnahan Nov 19 2011 at 3:12