(1) Apparently a general term of a sigma-field generated by a countable partition can be written down. For example, if $\mathcal{B} = \sigma(B_n,n\ge 1)$ and $\{B_n\}_{n\ge1}$ is a partition of the ground set $\Omega$, then a general element of $\mathcal{B}$ is of the form $\cup_{n \in I} B_n$ for some $I \subset \mathbb{N}$.
(2) Apparently, Borel $\sigma$-field (on $\mathbb{R}$) is countably generated (say by $\{(-\infty,q]:\; q\in \mathbb{Q}\}$) and I am told that there is no writing down such a generic formula for its elements.
- (1) seems to be a special case of a countably generated $\sigma$-field. Does this have a name? Can some more light be shed on the differences between this case and a more general countably generated $\sigma$-field? Or am I making some very obvious mistakes in the above statements?
Best Answer
If $(\Omega,\Sigma)$ is a countably generated measurable space, there is a natural partition of $\Omega$ into atoms, non-empty measurable sets that have no proper non-empty measurable subsets. Let $\mathcal{C}$ be a countable family such that $\sigma(\mathcal{C})=\Sigma$. Without loss of generality, we can assume that $\mathcal{C}$ is closed under complements. The atom containing $x$ is then exactly $$A(x)=\bigcap_{C\in\mathcal{C},x\in C}C.$$ Every measurable set is a union of atoms.
If the $\sigma$-algebra is generated by a countble partition, the atoms will be exactly the blocks of the partition. But a countably generated measurable space may have uncountably many atoms. For example, the real line with the Borel $\sigma$-field has the family of all singletons $\{r\}$ of real numbers $r$ as its atoms.
Now, every measurable set $B$ in a countably generated $\sigma$-algebra is a union of atoms. If there are only countably many atoms, every union of atoms will be a countable union and therefore measuable. But uncountable unions may not be measurable. If $N$ is not a Borel set, then it will still be a union of singletons and therefore a union of atoms.
So the general case of a countably generated measurable is more complicated because one cannot identify measurable sets with arbitrary unions of atoms.