Countably generated versus being generated by a countable partition

Question

(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?

score 5 · Accepted Answer · answered Feb 14 '13 at 11:22

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.

Thanks for the thorough response. I have heard atoms in the context of measures, and hadn't heard of them being used for $\sigma$-fields. Is this standard terminology? Do you have a reference which talks more about these? — passerby51, Feb 14 '13 at 12:31
A standard reference for this material is Borel Spaces by Rao and Rao. The term comes originally from Boolean algebra, and the meaning there applies directly. The term is not frequent, but standard. — Michael Greinecker, Feb 14 '13 at 12:58
Thanks. That one seems to be hard to find. There is no online version I suppose? and it seems to be out of print. — passerby51, Feb 14 '13 at 14:03
You can write me at [email protected] I can help you find access to a copy... — Michael Greinecker, Feb 14 '13 at 14:16
Thanks for the offer. I will think about how badly I want the book.... — passerby51, Feb 14 '13 at 23:11

Countably generated versus being generated by a countable partition

1 Answers1

Linked