= Solution
For <probability measures> on $(\Omega,\mathcal F)$, <absolute continuity of measures> means preservation of null sets in one direction:
$$
\boxed{\widetilde{\mathbb P}\ll\mathbb P\iff\bigl(\mathbb P(A)=0\Longrightarrow\widetilde{\mathbb P}(A)=0\bigr)\quad\text{for every }A\in\mathcal F.}
$$
Equivalently, by the <Radon-Nikodym theorem>, there is an $\mathcal F$-measurable <Radon-Nikodym derivative> $Z\geq0$ with $\mathbb E_{\mathbb P}Z=1$ and $\widetilde{\mathbb P}(A)=\mathbb E_{\mathbb P}[Z\mathbf1_A]$. 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.
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