Definition of An Infinite Sequence Being Random: Is the Infinite Monkey Theorem Sufficient?

Question

I realize that by the Infinity Monkey Theorem, an infinite sequence of numbers from 0 to 9 chosen randomly will contain every finite subsequence. My question: is this sufficient for a precise definition of an infinite sequence being random? That is, let $a_{k}$ $k$ a positive integer, being a sequence of integers between $0$ and $9$ (e.g. digits of a real number). Can we define that that is sequence will be random if for and $n$, and for all finite sequences $b_{j}(n)$ of number from $0$ to $9$ of length $n$ ($j$ ranges from $1$ to $n$), there exists an $m$ such that $a_{m-1+j}=b_{j}(n)$ for all $j$ ranging from $1$ to $n$. That is, $a_{k}$ is random precisely when the infinite sequence contains EVERY possible subsequence of EVERY possible length. Would this be a sufficient definition of randomness? If so, is it the criterion on which the randomness of the digits of $\pi$ rest? And has this been proven?

Answer 1

This is the tip of an alternative theory of probability which seems to have been formulated first by Richard von Mises, elaborated by Alonzo Church, and more recently by others, as listed in the cited Wikipedia article. See also this for an extended discussion.

This approach has had its ups and downs. Most mathematicians have, I think, rejected it in favor of the measure-theoretical formulation associated with Borel and Kolmogorov, at least as a useful starting point for day-to-day work. But I don't think there is universal agreement nowadays (by working probabilists and philosophers of probability) that such a theory is actually wrong.