$\newcommand\th\theta$Indeed, the densities $f_\th:=d\mu_\th/d\nu$ are defined only $\nu$-a.e., and hence it makes no sense to define a maximum likelihood estimate (MLE) as a maximizer of $f_\th(x)$ in $\th$ for whatever versions of the densities are chosen. 

Usually, though, it is assumed that versions $f_\th$ of the densities **can** be chosen so that the likelihood $L_x(\th):=f_\th(x)$ be regular enough in $\th$. In particular, in proofs of the [consistency][1] of the MLE, one of the standard assumptions is that the likelihood $L_x(\th)$ be continuous in $\th$; in proofs of [central limit theorems][2] for the MLE, one of the standard assumptions is that the likelihood $L_x(\th)$ be twice differentiable in $\th$.

For some textbook statistical models (that is, for some families $(\mu_\th)$), it is impossible to make $L_x(\th)$ be continuous in $\th$.  

   


  [1]: https://en.wikipedia.org/wiki/Maximum_likelihood_estimation#Consistency
  [2]: https://en.wikipedia.org/wiki/Maximum_likelihood_estimation#Efficiency