when is the maximum likelihood estimator measurable For a random variable $X$, a class of probability measures $P_\theta$ for $\theta\in \Theta$ and their densities $f_{\theta}$ w.r.t. some common measure $\mu$, we can define the maximum likelihood estimate:
$$\hat{\theta}^{*} = \hat{\theta}^{*} (X) = \arg
\max_{\theta \in \Theta} f_{\theta} (X)$$
In order for this to be a statistic, it needs to be measurable. When is it so? I can see the case where $\mu$ is the counting measure and $\Theta$ is finite in particular.
 A: It's an interesting question. Usually, a discussion on measurability of estimators is (deliberately) avoided in statistical applications...
In the following I assume the i.i.d. case; $(X,\mathcal{F},P)$ is a probability space, and $\Theta$ is a parameter space. The MLE is a special case of M-estimator $\hat\theta_n=\hat\theta_n(X_1,...,X_n)$ which is defined by
$$\tag{1}\label{1}
P_n h(\cdot,\hat\theta_n)=\inf_{\theta\in\Theta}P_nh(\cdot,\theta)
$$
for some measurable function $h:X\times\Theta\rightarrow \mathbb{R}$; ($P_n$ is the empirical measure/sample average). In the parametric likelihood case $h(x,\theta)=-\ln f_\theta(x)$.
The main issue with measurability of $\hat\theta_n$ is the measurability of the infimum in \eqref{1}. In particular, if $\Theta$ is compact and $h$ is continuous in $\theta$ for each $x\in X$, then there is enough structure to ensure that the relevant infimum is measurable (compactness implies separability so that we can consider the infimum over a countable dense subset of $\Theta$).
I found a more general discussion on the issue in "High Dimensional Probability, Vol. 1" on pages 34-58. First, let $\hat\theta_n^*$ denote an approximate M-estimator. A sequence of approximate estimators satisfies
$$P_nh(\cdot,\hat\theta_n^{*})-\inf_{\theta\in\Theta}P_nh(\cdot,\theta)\to 0$$
in (outer) probability (or a.s.). Imposing some structure on $(\Theta,\mathcal{S})$ (measurable space associated with $\Theta$) it can be shown that although a Borel-measurable approximate estimator need not exist, this structure ensures the existence of universally measurable (u.m.) estimator. 
Here is Theorem A.2 on page 55 (which uses a version of the measurable selection theorem): 

If $(X,\mathcal{F})$ is any measurable space and $(\Theta,\mathcal{S})$ is Suslin, and if a sequence of approximate M-estimators exists, then such estimators can be chosen to be u.m.

You can find even more discussion on measurability in chapter 7 of "Stochastic Optimal Control: The Discrete-Time Case" (it's available online).
A: The book Mathematical Foundations of Infinite-Dimensional Statistical Models by Giné and Nickl contains the following result as instructive excercise 7.2.3:
Lemma. Let $(X,\mathcal{A})$ be a measurable space, $\Theta$ a compact metric space and $u:X\times\Theta \rightarrow \mathbb{R}$ a function that is measurable in the first argument for fixed $\theta \in \Theta$ and continuous in the second argument for fixed $x\in X$. Then there exists a Borel measurable function $\hat \theta: X\rightarrow \Theta$ such that $u(x,\hat \theta(x)) = \sup_\Theta u(x,\theta)$ for all $x\in X$.
The crux is to make the choice of estimator canonical in some way. This is easiest illustrated for $\Theta\subset \mathbb{R}$, in which case one can always choose the largest maximiser. To show measurability one can then take a sequence $F_n\subset \Theta$ of finite subsets such that $\bigcup_n F_n\subset \Theta$ is dense and define $\hat \theta_n(x)$ to be the largest maximiser of $u(x,\cdot)\vert_{F_n}$. This is clearly measurable and hence $\hat \theta = \sup_n\hat \theta_n $ is measurable. 
