 Solution

On the same measurable space, absolute continuity of measures meansThe Radon-Nikodym theorem then supplies a nonnegative measurable density $Z$ with ${\mathbb{E}}_{\mathbb{P}}Z=1$ and $\tilde{\mathbb{P}}(A)={\mathbb{E}}_{\mathbb{P}}[Z{1}_A]$. The converse follows directly from this formula. Strict positivity of the Radon-Nikodym derivative gives equivalence of the two probability measures; absolute continuity of measures alone need not give equivalence.