I'm reading Deep Learning Architectures: A Mathematical Approach by Ovidiu Calin, and I'm trying to understand what exactly is a Baire measure by this definition:
Example C.3.6 (Baire measure) Let $K \subseteq \mathbb{R}^n$ and denote by $C^0(K)$ the set of all continuous real-valued functions with compact support (which vanish outside of a compact subset of $K$). The class of Baire sets, $B$, is defined to be the $\mathfrak{S}$-algebra generated by $\{x;f(x) \geq a\}$, with $f \in C^0(K)$. A Baire measure is a measure defined on $B$, such that $\mu(C) < \infty$, for all compact subsets $C \subset K$. It is worth noting that for $K \subseteq \mathbb{R}^n$ the class of Baire sets is the same as the class of Borel sets. In particular, any finite Borel measure is a Baire measure.
To me at least this definition is kind of hard to understand. Trying to break it down:
So we start off with $K$, a subset of $\mathbb{R}^n$, and then consider $C^0(K)=\{f_0,f_1,\ldots\}$, the set of all functions where each $f_i:\mathbb{R}^n\to\mathbb{R}$ is a function with compact support. In other words, if $x\not\in supp(f_i)$, then $x=0$ $\color{red}{(?_1)}$.
Alright, then for all $f_i\in C^0(K)$, consider all $x$ such that $f_i(x)\geq a$ for some $a\in\mathbb{R}$. The class of Baire sets $B$ is then the $\mathfrak{S}$-algebra generated by the set of all such $x$'s.
Finally, a Baire measure is just a measure defined on $B$ such that $\mu(C)<\infty$ for all compact subsets $C\subset K$.$\color{red}{(?_2)}$
$\color{red}{(?_1)}:$ is this correct?
$\color{red}{(?_2)}:$ What exactly does a compact subset mean here since $K\subset\mathbb{R}^n$? if $K\subset \mathbb{R}$ for example, a compact subset should just be all closed and bounded subsets, e.g. $[0,1]$. what is a compact subset in $\mathbb{R}^n$?
Finally, could anyone provide some examples/intuition as what a Baire measure is used for?
I can at least partially answer your questions.
I think your supposition that $C^0(K)$ is countable is not well-founded. There are uncountably many continuous, real-valued functions with support $K = [0, 1]$, for example.
By the Heine-Borel theorem, $K \subset \mathbb R^n$ being compact is exactly the same as $K$ being closed and bounded.
Your understanding of a Baire measure looks correct.