This is inspired by Carl Offner's reply to one of my previous questions and my previous question about marginal and joint measures.
- Given a measure $\mu$ on product $\sigma$-algebra $\prod_{i \in I} \mathbb{S}_i$ of a collection of measurable spaces $(X_i, \mathbb{S}_i), i \in I$, does there exist a measure $\mu_i$ on each component $\sigma$-algebra $\mathbb{S}_i$, s.t. their product $\prod_{i \in I} \mu_i$ is the given measure $\mu$ on the product $\sigma$-algebra?
- If no, what are some necessary and/or sufficient conditions for the given measure $\mu$ to have such a decomposition?
When they exist, how to construct the component measures $\mu_i$ from $\mu$?
For example, is this a viable way by defining $\mu_i(A_i):= \frac{\mu(A_i \times \prod_{j \in I, j\neq i} X_i)}{\prod_{j \in I, j\neq i} \mu_j(X_i)}, \forall A_i \in \mathbb{S}_i ?$ If not, when will it become viable? ADDED: I asked this question, because obviously, the product and the division may not make sense in some cases. Also I actually made a mistake of circular definition, where I define $\mu_i$ in terms of $\mu_j, j\neq i$ which have to be defined in similar ways.
Thanks and regards!