Strassen's theorem states that a necessary and sufficient condition for existence of a discrete-time martingale with a finite number $n$ of given marginals $\mu_1,\ldots,\mu_n$ is that the marginals increase in convex order.

Strassen's original proof (Theorem 8, https://projecteuclid.org/euclid.aoms/1177700153) shows this is the case for the 2 dimensions, I have no problem with this proof. Strassen then says if we know this holds for 2 dimensions, then the general case can be constructed as a Markov process.

I understand this has something to do with decomposition of measures, but I'm not too familiar with the theory.

How exactly are the measures glued together/combined whilst maintaining the martingale property? Where could I read more about the relevant theory?



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