Although -algebras are often introduced as a mere technicality in the definition of measurable space (to specify the measurable subsets), even once one has a fixed measurable space , it is often useful to consider additional (typically coarser) -algebras of measurable subsets of .
Notice that the power set of is a Boolean algebra under the operations of complementation and of union and intersection of finite families. Actually, it is a complete Boolean algebra, since we can also take unions and intersections of all families. A -algebra is an intermediate notion, since (in addition to being closed under complementation) we require that it be closed under unions and intersections of countable families.
It follows that is closed under intersections of inhabited finite families and under symmetric difference of finite families: * If and are in , then so is their intersection . * If and are in , then so is their symmetric difference .
We can actually use the latter as an alternative to (2), since . Or we can use the pair as an alternative to (2,3), since . For that matter, we can weaken (1) to simply say that some set is in ; then .
While the union and symmetric difference of an empty family (both the empty set) belong to , the intersection of an empty family (which is itself) might not. The term ‘ring’ dates from the days when a ring in algebra was not assumed to be unital; so a ring on is simply a subring (in this sense) of the Boolean ring .
A -ring on is a ring (as above) which is closed under intersections of countably infinite families. That is:
Of course, every -ring is a ring, but not conversely. Actually, if you want to define the concept of -ring directly, it's quicker if you use the symmetric difference; then (2,3) follow by the reasoning above and the idempotence of intersection (so that ).
The symbol ‘’ here is from German ‘Durchschnitt’, meaning intersection; it may be used in many contexts to refer to intersections of countable families.
A -ring on is a ring (as above) which is closed under unions of countably infinite families. That is:
Now (2) is simply redundant; . A -ring is obviously a ring, but in fact it is also a -ring; .
The symbol ‘’ here is from German ‘Summe’, meaning union; it may be used in many contexts to refer to unions of countable families.
An algebra or field on is a ring (as above) to which itself belongs. That is:
Actually, (2) is now redundant again; . But perhaps more importantly, is closed under absolute complementation (that is, complementation relative to the entire ambient set ); that is:
In light of this, the most common definition of algebra is probably to use this fact together with (1,2); then (3) follows because and (4) follows because . On the other hand, one could equally well use intersection instead of union; absolute complements allow the full use of de Morgan duality.
The term ‘field’ here is even more archaic than the term ‘ring’ above; indeed the only field in this sense which is a field (in the usual sense) under symmetric difference and intersection is the field (for an inhabited set ).
Finally, a -algebra or -field on is a ring that is both an algebra (as above) and a -ring (as above). That is:
As with -rings, (2) is redundant; as with algebras, it's probably most common to use the absolute complement in place of (3,4). Thus the usual definition of a -algebra states: 1. The empty set is in . 2. If is in , then so is its complement . 3. If are in , then so is their union .
And again we could again just as easily use intersection as union, even in the infinitary axiom. That is, a -algebra is automatically a -algebra, because .
Any and all of the above notions have been used by various authors in the definition of measurable space; for example, Kolmogorov used algebras (at least at first), and Halmos used -rings. Of course, the finitary notions (ring and algebra) aren't strong enough to describe the interesting features of Lebesgue measure; they are usually used to study very different examples (finitely additive measures). On the other hand, ‑ or -rings may be more convenient than -algebras for some purposes; for example, vector-valued measures on -rings make good sense even when the absolute measure of the whole space is infinite.
Note that the collection of measurable sets with finite measure (in a given measure space) is a -ring, while the collection of measurable sets with -finite measure is a -ring.
In any case, an -measurable subset of , or just a measurable set, is any subset of that belongs to . If is one of the more general variations, then we also want some subsidiary notions: is relatively measurable if belongs to whenever does, and is -measurable if it is a countable union of elements of . Notice that every relatively measurable set is measurable iff is at least an algebra; in any case, the relatively measurable sets form a ()-algebra if is at least a ()-ring. Similary, every -measurable set is measurable iff is at least a -ring; in any case, the -measurable sets form a -ring if is at least a -ring.
As a -algebra is a collection of subsets, we might hope to develop a theory of bases and subbases of -algebras, such as is done for topologies and uniformities. However, things do not work out as nicely. (It is quite easy to generate rings or algebras, but generating -rings and -rings is just as tricky as generating -algebras.)
We do get something by general abstract nonsense, of course. It's easy to see that the intersection of any collection of -algebras is itself a -algebra; that is, we have a Moore closure. So given any collection of sets whatsoever, the intersection of all -algebras containing is a -algebra, the -algebra generated by . (We can similarly define the -ring generated by and similar concepts for all of the other notions defined above.)
What is missing is a simple description of the -algebra generated by . (For a mere algebra, this is easy; any can be taken as a subbase of an algebra, the symmetric unions of finite families of elements of form a base of the algebra, and the intersections of finite families of elements of the base form an algebra. For a ring, the only difference is to use intersections only of inhabited families. But for anything from a -ring to a -algebra, nothing this simple will work.)
In fact, the question of how to generate a -algebra is the beginning of an entire field of mathematics, descriptive set theory?. For our purposes, we need this much:
So we need an steps, not just .
(This is only the beginning of descriptive set theory; our are their —except that for some reason they start with instead of —, and the subject continues to higher values of the superscript.)
Note that countable choice is essential here and elsewhere in measure theory, to show that a countable union of a countable union is a countable union. But the full axiom of choice is not; in fact, much of descriptive set theory (although this is irrelevant to the small portion above) works better with the axiom of determinacy instead.
Of course, the power set of is closed under all operations, so it is a -algebra.
If is a topological space, the -algebra generated by the open sets (or equivalently, by the closed sets) in is the Borel -algebra; its elements are called the Borel sets of . In particular, the Borel sets of real numbers are the Borel sets in the real line with its usual topology.
In the application of statistical physics to thermodynamics, we have both a microcanonical phase space (typically something like for on the order of Avogadro's number) describing every last detail of a physical system and a macrocanonical phase space (typically or at least for ) describing those features of the system that can be measured practically, with a projection . Then the preimage under this projection of the Borel -algebra of is a -algebra on , and the thermodynamic entropy of the system is (theoretically) its information-theoretic entropy with respect to this -algebra.
If a measurable space is equipped with a (positive) measure , making it into a measure space, then the sets of measure zero form a -ideal of (that is an ideal that is also a sub--ring). Let a null set be any set (measurable or not) contained in a set of measure zero; then the null sets form a -ideal in the power set of . Call a set -measurable if it is the union of a measurable set and a null set; then the -measurable sets form a -algebra called the completion of under . (Even if is only a -ring, still the null sets will form a -ring; in any case, we get as completion the same kind of structure as we began with.) Note that we can also do this by starting with any -ideal and simply declaring by fiat that these are the null sets, as with a localisable measurable space; then we speak of the completion of with respect to (or sometimes with respect to the -filter of full sets).
In particular, the Lebesgue-measurable sets in the real line are the elements of the completion of the Borel -algebra under Lebesgue measure.
We are now learning ways to understand measure theory and probability away from the traditional reliance on sets required with -algebras; see measurable space for a summary of other ways to define this concept. We still need to know what happens to all of the other -algebras of measurable sets in a measurable space. One solution may to use quotient measurable spaces in place of sub--algebras; for example, see explicit quotient in the example of macroscopic entropy above.