Defining fractals in a probability space
Chaoshou Dai, Samuel J. Taylor · Illinois Journal of Mathematics · 1994
The Billingsley formulation (which we use) depends critically on a fixed stochastic process {X, X2,... taking values in a finite or countable state space S. No information about a subset E c f has any relevance unless it depends on the values {Xn(to)}.This means that we would lose nothing by assuming that f is an infinite product of a sequence of copies of S, which would make a typical point to (Xl, x2,..., xn,...), where each x S.This idea is developed and used extensively by Cajar [3].We keep it in mind and note that it is the reason why so many of our arguments are simpler than in the Euclidean case.However, we find it more intuitive to build on the original formulation and notation of Billingsley.