Lemma 2 on page 637 of Jennrich (1967) states that:
Let $Q$ be a real-valued function on $\Theta\times Y$ where $\Theta$ is a compact subset of a Euclidean space and $Y$ is a measurable space. For each $\theta$ in $\Theta$ let $Q(\theta,y)$ be a measurable function of $y$ and for each $y$ in $Y$ a continuous function of $\theta$. Then there exists a measurable function $\hat{\theta}$ from $Y$ into $\Theta$ such that for all $y$ in $Y$: $$ Q(\hat{\theta}(y),y)=\inf_\theta Q(\theta,y). $$
I have a few questions about this please:
- Where can I find an alternative source (preferably a text book) of proof for this result? Jennrich's writing is too dense for me at the moment. If you know a relatively accessible proof, please share it here with me and others.
- Jennrich's proof begins with:
Let $\{\Theta_n\}$ be an increasing sequence of finite subsets of $\Theta$ whose limit is dense in $\Theta$.
This seems to be a consequence of $\Theta$ being compact. What is the name of this property?
Thank you!