TITLE: when is the maximum likelihood estimator measurable
QUESTION [10 upvotes]: 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.

REPLY [2 votes]: 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.