Suppose that we have a parameter space $\Theta$ that is NOT compact.
The M-estimator is defined to be $\widehat{\theta}_{n}$ which maximizes $M_{n}\left(\theta\right)=\sum_{i=1}^{n}m_{\theta}\left(X_{i}\right)$ and $\theta^{*}$ maximizes $M\left(\theta\right)=\mathbb{E}\left[m_{\theta}\left(X\right)\right]$, for some functions $m_{\theta}$ and $X_{1},\ldots,X_{n}$ random variables i.i.d. from a pdf $f$.
The expectation $\mathbb{E}$ is with respect to $f$. Assume that there exists a compact set $S\in \Theta$ such that $\theta^{*}\in X$ and
\begin{equation}
\mathbb{E}\left[\sup_{\theta\in\mathcal{\Theta}\cap S^{c}}m_{\theta}\left(X\right)\right]