$\sigma$-algebras and their variants are collections of subsets important in classical measure theory and probability theory.
Although $\sigma$-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 $X$, it is often useful to consider additional (typically coarser) $\sigma$-algebras of measurable subsets of $X$.
We assume the law of excluded middle throughout; see Cheng measurable space for the constructive theory, and compare also measurable locale.
Given a set $X$, a $\sigma$-algebra is a collection of subsets of $X$ that is closed under complementation and under unions and intersections of countable families.
Notice that the power set $\mathcal{P} X$ of $X$ 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 $\sigma$-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.
Given a set $X$ and a collection $\mathcal{M}$ of subsets $S \subseteq X$, there are really several kinds of collections that $\mathcal{M}$ could be:
A ring on $X$ is a collection $\mathcal{M}$ which is closed under relative complementation and under unions of finitary families. That is:
It follows that $\mathcal{M}$ is closed under intersections of inhabited finite families and under symmetric difference of finite families:
We can actually use the latter as an alternative to (2), since $S \cup T = (S \uplus T) \uplus (S \cap T)$. Or we can use the pair as an alternative to (2,3), since $T \setminus S = (S \cap T) \uplus T$. For that matter, we can weaken (1) to simply say that some set $S$ is in $\mathcal{M}$; then $\empty = S \setminus S$.
While the union and symmetric difference of an empty family (both the empty set) belong to $\mathcal{M}$, the intersection of an empty family (which is $S$ 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 $X$ is simply a subring (in this sense) of the Boolean ring $\mathcal{P} X$.
A $\delta$-ring on $X$ is a ring (as above) $\mathcal{M}$ which is closed under intersections of countably infinite families. That is:
Of course, every $\delta$-ring is a ring, but not conversely. Actually, if you want to define the concept of $\delta$-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 $S \cap T = S \cap T \cap T \cap T \cap \cdots$).
The symbol ‘$\delta$’ here is from German ‘Durchschnitt’, meaning intersection; it may be used in many contexts to refer to intersections of countable families.
A $\sigma$-ring on $X$ is a ring (as above) $\mathcal{M}$ which is closed under unions of countably infinite families. That is:
Now (2) is simply redundant; $S \cup T = S \cup T \cup T \cup T \cup \cdots$. A $\sigma$-ring is obviously a ring, but in fact it is also a $\delta$-ring; $\bigcap_i S_i = (\bigcup_i S_i) \setminus \bigcup_j (\bigcup_i S_i \setminus S_j)$.
The symbol ‘$\sigma$’ 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 $X$ is a ring (as above) $\mathcal{M}$ to which $X$ itself belongs. That is:
Actually, (2) is now redundant again; $S \cup T = X \setminus ((X \setminus T) \setminus S)$. But perhaps more importantly, $\mathcal{M}$ is closed under absolute complementation (that is, complementation relative to the entire ambient set $X$); 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 $T \setminus S = \neg(S \cup \neg{T})$ and (4) follows because $X = \neg\empty$. 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 $\{\empty, X\}$ (for an inhabited set $X$).
Finally, a $\sigma$-algebra or $\sigma$-field on $X$ is a ring $\mathcal{M}$ that is both an algebra (as above) and a $\sigma$-ring (as above). That is:
As with $\sigma$-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 $\sigma$-algebra states:
And again we could again just as easily use intersection as union, even in the infinitary axiom. That is, a $\delta$-algebra is automatically a $\sigma$-algebra, because $\bigcup_i S_i = \neg\bigcap_i \neg{S_i}$.
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 $\sigma$-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, $\delta$‑ or $\sigma$-rings may be more convenient than $\sigma$-algebras for some purposes; for example, vector-valued measures on $\delta$-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 $\delta$-ring, while the collection of measurable sets with $\sigma$-finite measure is a $\sigma$-ring.
A measurable space is usually defined to be a set $X$ with a $\sigma$-algebra $\mathcal{M}$ on $X$; sometimes one of the more general variations above is used.
In any case, an $\mathcal{M}$-measurable subset of $X$, or just a measurable set, is any subset of $X$ that belongs to $\mathcal{M}$. If $\mathcal{M}$ is one of the more general variations, then we also want some subsidiary notions: $S$ is relatively measurable if $S \cap T$ belongs to $\mathcal{M}$ whenever $T$ does, and $S$ is $\sigma$-measurable if it is a countable union of elements of $\mathcal{M}$. Notice that every relatively measurable set is measurable iff $S$ is at least an algebra; in any case, the relatively measurable sets form a ($\sigma$)-algebra if $\mathcal{M}$ is at least a ($\delta$)-ring. Similary, every $\sigma$-measurable set is measurable iff $S$ is at least a $\sigma$-ring; in any case, the $\sigma$-measurable sets form a $\sigma$-ring if $\mathcal{M}$ is at least a $\delta$-ring.
As a $\sigma$-algebra is a collection of subsets, we might hope to develop a theory of bases and subbases of $\sigma$-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 $\delta$-rings and $\sigma$-rings is just as tricky as generating $\sigma$-algebras.)
We do get something by general abstract nonsense, of course. It's easy to see that the intersection of any collection of $\sigma$-algebras is itself a $\sigma$-algebra; that is, we have a Moore closure. So given any collection $\mathcal{B}$ of sets whatsoever, the intersection of all $\sigma$-algebras containing $\mathcal{B}$ is a $\sigma$-algebra, the $\sigma$-algebra generated by $\mathcal{B}$. (We can similarly define the $\delta$-ring generated by $\mathcal{B}$ and similar concepts for all of the other notions defined above.)
What is missing is a simple description of the $\sigma$-algebra generated by $\mathcal{B}$. (For a mere algebra, this is easy; any $\mathcal{B}$ can be taken as a subbase of an algebra, the symmetric unions of finite families of elements of $\mathcal{B}$ 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 $\delta$-ring to a $\sigma$-algebra, nothing this simple will work.)
In fact, the question of how to generate a $\sigma$-algebra is the beginning of an entire field of mathematics, descriptive set theory?. For our purposes, we need this much:
So we need an $\aleph_1$ steps, not just $2$.
(This is only the beginning of descriptive set theory; our $\Sigma_\alpha$ are their $\Sigma^0_\alpha$ —except that for some reason they start with $\Sigma^0_1$ instead of $\Sigma^0_0$—, 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 $X$ is closed under all operations, so it is a $\sigma$-algebra.
If $X$ is a topological space, the $\sigma$-algebra generated by the open sets (or equivalently, by the closed sets) in $X$ is the Borel $\sigma$-algebra; its elements are called the Borel sets of $X$. 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 $P$ (typically something like $\mathbb{R}^N$ for $N$ on the order of Avogadro's number) describing every last detail of a physical system and a macrocanonical phase space $p$ (typically $\mathbb{R}^2$ or at least $\mathbb{R}^n$ for $n \lt 10$) describing those features of the system that can be measured practically, with a projection $P \to p$. Then the preimage under this projection of the Borel $\sigma$-algebra of $p$ is a $\sigma$-algebra on $P$, and the thermodynamic entropy of the system is (theoretically) its information-theoretic entropy with respect to this $\sigma$-algebra.
If a measurable space $(X,\mathcal{M})$ is equipped with a (positive) measure $\mu$, making it into a measure space, then the sets of measure zero form a $\sigma$-ideal of $\mathcal{M}$ (that is an ideal that is also a sub-$\sigma$-ring). Let a null set be any set (measurable or not) contained in a set of measure zero; then the null sets form a $\sigma$-ideal in the power set of $X$. Call a set $\mu$-measurable if it is the union of a measurable set and a null set; then the $\mu$-measurable sets form a $\sigma$-algebra called the completion of $\mathcal{M}$ under $\mu$. (Even if $\mathcal{M}$ is only a $\delta$-ring, still the null sets will form a $\sigma$-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 $\sigma$-ideal $\mathcal{N}$ and simply declaring by fiat that these are the null sets, as with a localisable measurable space; then we speak of the completion of $\mathcal{M}$ with respect to $\mathcal{N}$ (or sometimes with respect to the $\delta$-filter $\mathcal{F}$ of full sets).
In particular, the Lebesgue-measurable sets in the real line are the elements of the completion of the Borel $\sigma$-algebra under Lebesgue measure.
We are now learning ways to understand measure theory and probability away from the traditional reliance on sets required with $\sigma$-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 $\sigma$-algebras of measurable sets in a measurable space. One solution may to use quotient measurable spaces in place of sub-$\sigma$-algebras; for example, see explicit quotient in the example of macroscopic entropy above.