Measurable Partitions

Question

I was wondering in the definition of the Measurable Partition what does the notation ; $\bigvee_{n=1}^\infty P_n $ means? We say that $P$ is a measurable partition if there exist some measurable set $M_0 \subset M$ with full measure such that, restricted to $M_0$, $$p=\bigvee_{n=1}^\infty P_n$$ for some increasing sequence $p_1 \prec p_2 ... \prec p_n\prec ... $ of countable partitions. By $p_i \prec p_{i+1}$ we mean that every element of $p_{i+1}$ is contained in some elements of $p_i$.

Answer 1

You mention that you are reading Foundations of Ergodic Theory, I suppose the book written by Viana and Oliveira. The definition of measurable partition you give is on page 146 of the english version of the book (147 of the Portuguese version available online). Immediately after that definition, the authors define the notation $$ \bigvee_{n=1}^\infty \mathcal{P}_n$$

"Represent by $\bigvee_{n=1}^\infty \mathcal{P}_n$ the partition whose elements are the non-empty intersections of the form $\bigcap_{n=1}^\infty {P}_n$ with $P_n\in \mathcal{P}_n$ for every $n$. Equivalently, this is the coarser partition such that $$\mathcal{P}_n \prec \bigvee_{n=1}^\infty \mathcal{P}_n $$ for every $n$"

The symbol $\prec$ is defined in the previous paragraph: "By $\mathcal{P_i}\prec\mathcal{P}_{i+1}$ we mean that every element of $\mathcal{P}_{i+1}$ is contained in some element of $\mathcal{P}_{i}$."

Answer 2

It doesn't denote a measurable partition. Instead $\bigvee_{n=1}^\infty P_n$ denotes the smallest $\sigma$-algebra containing all intersections $\bigcap_{n=1}^\infty A_n$ with $A_n\in P_n$ for each $n$.

However, in Rokhlin's notion of a measurable partition, the notation $\bigvee_{n=1}^\infty P_n$ denotes the collection of all sets of the form $\bigcap_{n=1}^\infty A_n$ with $A_n\in P_n$ for each $n$.

In order to understand the difference between the two take the collection of stable invariant manifolds in a well-behaved system. Often this is an uncountable partition unlike in the "usual" notion of a measurable partition. In many situations this is a measurable partition in the sense of Rokhlin.