You need the Radon-Nikodym theorem.

In the measure-theoretic problem, you are estimating a distribution $F_0$ that is defined on the sigma-algebra on the sample set, not on the sample set itself. Now suppose you have a parametric model $F(\cdot|\theta)$. If this model is dominated by a $\sigma$-finite measure $\mu$, then (by Radon-Nikodym) we can safely move to the associated family of density fuctions $f(\cdot|\theta)$, satisfying $F(A|\theta) = \int_A f(y|\theta) d\mu(y)$.

Clearly, if your measurement lies in a null set for every distribution in your model, your model or your observation is wrong (since the probability of your observation is zero in all distribution that are considered). On the other hand, if there is a distribution for which your measurement does not lie in any null set, then it also can't lie in a null set of $\mu$, and so you are not free to redefine your densities at your observation.

Note, however, that existence nor uniqueness of the MLE are guaranteed.

You need the Radon–Nikodym theorem..

In the measure-theoretic problem, you are estimating a distribution $F_0$ that is defined on the sigma-algebra on the sample set, not on the sample set itself. Now suppose you have a parametric model $F(\cdot\mid\theta)$. If this model is dominated by a $\sigma$-finite measure $\mu$, then (by Radon–Nikodym) we can safely move to the associated family of density functions $f(\cdot \mid \theta)$, satisfying $F(A\mid\theta) = \int_A f(y\mid\theta) \, d\mu(y)$.

Clearly, if your measurement lies in a null set for every distribution in your model, your model or your observation is wrong (since the probability of your observation is zero in all distribution that are considered). On the other hand, if there is a distribution for which your measurement does not lie in any null set, then it also can't lie in a null set of $\mu$, and so you are not free to redefine your densities at your observation.

Note, however, that existence nor uniqueness of the MLE are guaranteed.