 Solution

For probability measures on $(\Omega ,\mathcal{F})$, absolute continuity of measures means preservation of null sets in one direction:Equivalently, by the Radon-Nikodym theorem, there is an $\mathcal{F}$-measurable Radon-Nikodym derivative $Z\ge 0$ with ${\mathbb{E}}_{\mathbb{P}}Z=1$ and $\tilde{\mathbb{P}}(A)={\mathbb{E}}_{\mathbb{P}}[Z{1}_A]$. The derivative may vanish, so absolute continuity of measures does not require equivalence of the two probability measures. In particular, the reverse null-set implication is not part of the definition.