# Probability: Information contained in sigma-algebras

Consider a sample space $\Omega$ and a sigma-algebra $F$ over it. Let $A$ be a class of $F$-sets. $A$ then generates a partition of $\Omega$ based on the following equivalence relation: Points $w_{1}$ and $w_{2}$ of $\Omega$ are related if, for every set $B$ in $A$, they both belong or they both don't belong to $B$ [ref: Billingsley, 3rd edition, Chapter 1, Section 4, Subfields]. Further, this partition is the same as that generated by the sigma-algebra of $A$ [$\sigma(A)$] defined using the same relation.

Q1. Does it mean that $\sigma(A)$ gives exactly the same information that $A$ gives? In other words, is knowing either $A$ or its sigma-algebra enough to extract any information that we could possibly seek?

Q2. Is there any condition for the above to hold? For example, if $A$ is not a pi-system (not closed under finite intersections) does it hold?

Q3. Can independence of 2 classes $C$ and $D$ (or the lack of it) be inferred by studying their sigma-algebras? For example, if a set in $D$ lies in $\sigma(C)$, is it enough to ensure that $C$ and $D$ are not independent? If so, why?

-

Q3. Independence is related to what probability measure you are using. So I'd say it cannot be inferred from the sigma-algebras. Regarding your example $\Omega$ is always in $\sigma(C)$, but it can also be in an independent sigma algebra.
Q1: The answer is no. Let $\Omega=[0,1]$. Let $A_1$ be the Borel $\sigma$-algebra on $\Omega$ and let $A_2$ be the $\sigma$-algebra consisting of subsets of $\Omega$ that are countable or have a countable complement. Both lead to the partition into singletons, yet are quite different. In particular, a $A_2$-measurable real valued function is constant on a set with Lebesgue measure $1$. To see this, let $f:\Omega\to\mathbb{R}$ be $A_2$-measurable. Note that for each $n$, there is a set of the form $[z/2^n,(z+1)/2^n]$ for some $z\in\mathbb{Z}$ such that $\mu f^{-1}\big([z/2^n,(z+1)/2^n]\big)=1$. Picking such a set for each $n$ and taking their countable intersection, you get a singleton set $\{x\}$ such that $\mu f^{-1}(\{x\})=1$.
Q2: There is a class of measurable spaces for which the answer is almost yes. A countably generated measurable space in which all points are measurable is strongly Blackwell if any two countably generated sub-$\sigma$-algebras that generate the same partition are equal. David Blackwell has shown (corollary 1) that any analytic set is strongly Blackwell.