Question: For example, if I took the first billion digits of pi, in order, and then chopped those digits up into an array of, let's say 4 character chunks, would those 250M numbers appear random? By "appear random" I mean when run through the standard statistical measures of statistical randomness.

Answer: 
There is no reason to think that $$\lim\_{n\to\infty}\mu\left(\bigcup\_{k=1}^n A\_k\right) = \mu\left(\bigcup\_{k=1}^\infty A\_k\right)$$

It's clear that, with finite additivity, you get $$\lim\_{n\to\infty}\mu\left(\bigcup\_{k=1}^n A\_k\right) \leq \mu\left(\bigcup\_{k=1}^\infty A\_k\right)$$

but you want equality.

A simple example where that would fail is the natural number "density" measure. For $X\subseteq \mathbb N$, define $$\mu(X)=\lim\_{n\to\infty} \frac{\left|X\cap [1,n]\right|}{n}$$

This satisfies finite additivity, but not countable additivity.

For example, $A\_k=\{k\}$ each have measure zero, but $\mu\left(\cup A\_k\right)=1$.
What you are talking about is called a **finitely additive measure**. There are finitely additive measures that are not proper measures (*i.e.*, not countably additive). See [here](http://en.wikipedia.org/wiki/Sigma_additivity#An_additive_function_which_is_not_.CF.83-additive) for a counterexample.