Bernoulli Process - As A Metric Space

As A Metric Space

Given any two infinite binary sequences and, one can define a metric, and, in fact, an ultrametric by considering the first location where these two strings differ. That is, let

One then defines the distance between x and y as

This metric is known as the k-adic metric (for k=2). With it, the Bernoulli process becomes a compact metric space. The metric topology induced by this metric results in exactly the same Borel sigma algebra as that constructed from the cylinder sets; this is essentially because the open balls induced by the metric are complements of the cylinder sets (the only points in are the infinite strings).