What is Measure on function space? I am having trouble understanding what is measure on a function space. Taylor in Introduction to Measure and Integration begins by defining  a  product space $\prod\limits_{i\in I}^{}X_i$ and a set of function on the index $f:I\to\bigcup\limits_{i\in I}X_i$ in which $f(i)\in X_i$ "In the particular case where $X_i$ is the same space $X$ for all $I$, the space $\prod\limits_{i\in I}^{}X_i$ reduces to the set of functions: $I\to X$" Could someone explain me what does it mean to have a set of function whose domain is index? What does it mean a point in$\prod\limits_{i\in I}^{}X_i$ can be thought as a function $f:I\to\bigcup\limits_{i\in I}X_i$?  What is the relation to a measure on function space? Does the Kolmogorov extension theorem has to do with this? Thanks for reading!
 A: The statements are all about how to generalize the definition of a product space $X_1\times X_2$ to the product of arbitrarily many spaces. Functions appear in this context as the easiest way to index a huge collection of components.
For finite products, the usual way to denote points in the space $X_1\times X_2\times\dots\times X_n$ is $(x_1,x_2,\dots,x_n)$, where each single point $x_j$ should be contained in the corresponding space $X_j$. 
For countable products $\prod_{k=0}^{\infty}X_k=X_0\times X_1\times X_2\times\dots$ the natural extension of the above is to consider infinite sequences $(x_k)_{k=0}^\infty=(x_0,x_1,x_2,\dots)$, where again each element should be in the corresponding space: $x_k\in X_k$. Another way to view this is by seeing the sequence $(x_k)$ as a map
$$x:\mathbb{N}\to \bigcup_{k=0}^\infty X_k,\quad \text{where }x(k)\in X_k.$$
The nice part with this latter formulation is that now it really doesn't matter how big (or small) our collection of indices is. For any collection of indices $I$, we can use the above idea to define the product space $\prod_{i\in I}X_i$ as the space whose points are themselves described by maps, that is,
$$\prod_{i\in I}X_i=\Big\{x:I\to\bigcup_{i\in I}X_i,\quad \text{where }x(i)\in X_i\text{ for all }i\in I\Big\}.$$
