Given two measure spaces $(\Omega_i, \mathcal{F}_i, \mu_i), i=1,2$, does there always exists a measure preserving mapping $(\Omega_1, \mathcal{F}_1, \mu_1) \to (\Omega_2, \mathcal{F}_2, \mu_2)$?
One necessary condition is that $\mu_1(\Omega_1) = \mu_2(\Omega_2)$. But is that also sufficient?
I am in the middle of understanding what the Kolmogorov extension theorem is all about.
- Some version states without a stochastic process, i.e. for any consistent finite dimensional distributions, under some condition, there exists a measure on the space of the sample paths s.t. the finite dimensional distributions are its marginal distributions.
- Some version states in terms of a stochastic process, i.e. for any consistent finite dimensional distributions, under some condition, there exists a stochastic process whose measure on the space of the sample paths has the finite dimensional distributions as its marginal distributions.
Obviously the latter version implies the former version, but I am not sure the former can imply the latter. That serves as the source of my questions.
BTW, are the measures in KET all probability measures, not general ones? I saw at the end of this note, the measures in its version of KET are required to be Baire measures, which I think may not be probability measures?
Thanks and regards!