# Can an arbitrary probability space be simulated by coin tosses?

Let $$B=\left(\{\mathrm{heads},\mathrm{tails}\},\mathcal{P}\{\mathrm{heads},\mathrm{tails}\},\mu\right)$$ be the probability space for a single fair coin toss. For any cardinal $$\aleph$$ let $$B^\aleph$$ be the independent product probability space, representing $$\aleph$$ independent fair coin tosses.

For each probability space $$A$$, is there some cardinal $$\aleph$$ such that there exists a measure-preserving measurable function $$B^\aleph\rightarrow A$$?

## This question has an open bounty worth +200 reputation from Oscar Cunningham ending in 4 days.

This question has not received enough attention.

• I sort of doubt that any continuous parameter stochastic processes could be modeled that way. – zoli Feb 10 '15 at 0:11
• It is so if $X$ is a Borel space with cardinal being countable (cf. Kallenberg, chapter 3), which covers most interesting cases, for instance, every Polish space will do. Dunno about the general case. – Jorkug Feb 10 '15 at 7:15
• @zoli You can certainally model the "sample continuous" stochastic processes with countably many coinflips, and indeed this is precisely how computers are able to simulate them. – Oscar Cunningham Feb 11 '15 at 11:41
• I'm thinking about the Dieudonné measure on $\omega_1$. It's not a very interesting space for probability, since every event has probability 0 or 1, but still I don't see how to get it as a pushforward of a coin flip measure. – Nate Eldredge Jul 14 at 14:55
• Essentially, the answer to the question is yes by Maharam's structure theorem. However, the measurable functions given by this are isomorphisms of the measure algebra and not defined point wise on the underlying set. – George Lowther 21 hours ago