I am trying to prove a version of Mercer's Theorem for an arbitrary compact metric space; that is, I do not wish to restrict myself to the space of real-valued continuous functions $C[a,b]$. I understand the proof for the case of $C[a,b]$, but for the arbitrary metric space case, I need to define integration in that arbitrary space.
In other words, given compact metric spaces $(X,d_X)$ and $(Y,d_Y)$, and let $f:X\rightarrow Y$. How can we define the integral
$$\displaystyle\int_X f(x)dx$$
so that it is well-defined no matter what our choice of $X$ and $Y$ is?
Are there any resources that explain this clearly and fully? I have a pretty solid understanding of basic functional analysis and some understanding of point-set topology. I think this is related to Measure Theory, but I'm not sure exactly how or where to start.
A simple example of why this is headed in a wrong direction: If $X=\mathbb [0,1]$ and $Y_1=[0,1]$, then the integral might be defined as normal. If $Y_2=[-1/2,1/2]$, then the integral defined as normal also exists. But, as metric spaces without knowing their "real number" structure, $Y_1$ and $Y_2$ are essentially the same metric space, but the integral returns a radically different value.
Essentially, there is not a canonical integral even in the case $X=[0,1]$ and $Y$ being isomorphic to $[0,1]$ as a metric space.
You need much more structure than merely being compact metric spaces.