Why is an M-estimator from statistics not necessarily measurable?

Question

I read somewhere that an M-estimator, defined as estimators that maximize a criterion of the form:

$$ \theta \mapsto M_n(\theta) = \sum_{i=1}^{n}m_\theta(Y_i) $$ for some functions $\{m_\theta: \theta \in \mathcal{H}\}$. $Y_i$ here are random variables.

I read a remark in a textbook that an M-estimator is not necessarily measurable, since it is defined as an argmin or an argmax over an uncountable set $\mathcal{H}$.

I do not understand what minimizing and maximizing over an uncountable set necessarily implies measurability. Could someone help me here? thanks.

Answer 1

First note that for all $\theta \in \mathcal{H}$, $\sum_{i=1}^n m_\theta(Y_i)$ is a measurable function, $$\sum_{i=1}^n m_\theta(Y_i): (\mathbb{R}^d, \mathcal{B(\mathbb{R}^d)}) \rightarrow (\mathbb{R},\mathcal{B(\mathbb{R})}),$$ say. To simplify notation instead write $f_\theta: (\mathbb{R}^d, \mathcal{B(\mathbb{R}^d)}) \rightarrow (\mathbb{R},\mathcal{B(\mathbb{R})})$, for all $\theta \in \mathcal{H}$.

First suppose that $\mathcal{H}$ is countable. Then $\operatorname{sup}_{\theta \in \mathcal{H}}f_\theta$ is measurable since $$\forall a \in \mathbb{R}: \{\operatorname{sup}_{\theta \in \mathcal{H}}f_\theta \leq a \} = \bigcap_{\theta \in \mathcal{H}}\{f_\theta \leq a\},$$ which is measurable as a countable intersection of measurable sets.

Now suppose that $\mathcal{H}$ can be more than countable. Let $\mathcal{H}$ be a non-measurable set and for all $\theta$ suppose that $f_\theta(x) = 1(x=\theta)$. Then each $f_\theta$ is measurable, but $\operatorname{sup}_{\theta \in \mathcal{H}}f_\theta = 1_\mathcal{H}$ is not measurable.

You can easily construct a similar counterexample if $\mathcal{H}$ is measurable but has a non-measurable subset.