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If you are given a finite fumber of r.v. say $X_1...X_n$ and you know say the conditional distribution of each one given the preceding ones (i.e with smaller index) values , and you also know the marginal say $X_1$ then there is a unique measure satisfying these conditions

I am curious wether this generalizes to an infinite family indexed say by $R$. The marginal distributions could be expressed given the sigma algebra generated by the preceding ones.

eternalstudent
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  • Hi: based on your question, I think you might be referring to the kolmogorov extension theorem. check that out because, if I'm correct, then you'll find a lot about it is discussed on this list in past threads. – mark leeds Jul 04 '22 at 16:27
  • That theorem only considers the special case that the marginals are independants, although it can "probably" be used as a lemma for this problem. – eternalstudent Jul 04 '22 at 16:32

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