Kolmogorov Extension Theorem - Statement of The Theorem

Statement of The Theorem

Let denote some interval (thought of as "time"), and let . For each and finite sequence of times, let be a probability measure on . Suppose that these measures satisfy two consistency conditions:

1. for all permutations of and measurable sets ,

2. for all measurable sets ,

Then there exists a probability space and a stochastic process such that

for all, and measurable sets, i.e. has as its finite-dimensional distributions relative to times .

In fact, it is always possible to take as the underlying probability space and to take for the canonical process . Therefore, an alternative way of stating Kolomogorov's extension theorem is that, provided that the above consistency conditions hold, there exists a (unique) measure on with marginals for any finite collection of times . The remarkable feature of Kolmogorov's extension theorem is that it does not require to be countable, but the price to pay for this level of generality is that the measure is only defined on the product σ-algebra of, which is not very rich.