nLab
sigma-algebra

Idea

$\sigma$-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 $X$.

Definitions

We assume the law of excluded middle throughout; see Cheng measurable space for the constructive theory, and compare also measurable locale.

Short version

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 $𝒫X$ 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.

Long version

Given a set $X$ and a collection $ℳ$ of subsets $S\subseteq X$, there are really several kinds of collections that $ℳ$ could be:

• A ring on $X$ is a collection $ℳ$ which is closed under relative complementation and under unions of finitary families. That is:

  1. The empty set $\varnothing$ is in $ℳ$.
  2. If $S$ and $T$ are in $ℳ$, then so is their union $S\cup T$.
  3. If $S$ and $T$ are in $ℳ$, then so is their relative complement $T\setminus S$.

  It follows that $ℳ$ is closed under intersections of inhabited finite families and under symmetric difference of finite families:

  • If $S$ and $T$ are in $ℳ$, then so is their intersection $S\cap T=T\setminus (T\setminus S)$.
  • If $S$ and $T$ are in $ℳ$, then so is their symmetric difference $S\uplus T=(T\setminus S)\cup (S\setminus T)$.

  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 $ℳ$; then $\varnothing =S\setminus S$.

  While the union and symmetric difference of an empty family (both the empty set) belong to $ℳ$, 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 $𝒫X$.

• A $\delta$-ring on $X$ is a ring (as above) $ℳ$ which is closed under intersections of countably infinite families. That is:

  1. The empty set $\varnothing$ is in $ℳ$.
  2. If $S$ and $T$ are in $ℳ$, then so is their union $S\cup T$.
  3. If $S$ and $T$ are in $ℳ$, then so is their relative complement $T\setminus S$.
  4. If $S_1,S_2,S_3,\dots$ are in $ℳ$, then so is their intersection ${\bigcap }_i{S}_i$.

  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) $ℳ$ which is closed under unions of countably infinite families. That is:

  1. The empty set $\varnothing$ is in $ℳ$.
  2. If $S$ and $T$ are in $ℳ$, then so is their union $S\cup T$.
  3. If $S$ and $T$ are in $ℳ$, then so is their relative complement $T\setminus S$.
  4. If $S_1,S_2,S_3,\dots$ are in $ℳ$, then so is their union ${\bigcup }_i{S}_i$.

  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) $ℳ$ to which $X$ itself belongs. That is:

  1. The empty set $\varnothing$ is in $ℳ$.
  2. If $S$ and $T$ are in $ℳ$, then so is their union $S\cup T$.
  3. If $S$ and $T$ are in $ℳ$, then so is their relative complement $T\setminus S$.
  4. The improper subset $X$ is in $ℳ$.

  Actually, (2) is now redundant again; $S\cup T=X\setminus ((X\setminus T)\setminus S)$. But perhaps more importantly, $ℳ$ is closed under absolute complementation (that is, complementation relative to the entire ambient set $X$); that is:

  • If $S$ is in $ℳ$, then so is its complement $\neg S$.

  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 $X=\neg \varnothing$. 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 $\{\varnothing ,X\}$ (for an inhabited set $X$).

• Finally, a $\sigma$-algebra or $\sigma$-field on $X$ is a ring $ℳ$ that is both an algebra (as above) and a $\sigma$-ring (as above). That is:

  1. The empty set $\varnothing$ is in $ℳ$.
  2. If $S$ and $T$ are in $ℳ$, then so is their union $S\cup T$.
  3. If $S$ and $T$ are in $ℳ$, then so is their relative complement $T\setminus S$.
  4. The improper subset $X$ is in $ℳ$.
  5. If $S_1,S_2,S_3,\dots$ are in $ℳ$, then so is their union ${\bigcup }_i{S}_i$.

  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:

  1. The empty set $\varnothing$ is in $ℳ$.
  2. If $S$ is in $ℳ$, then so is its complement $\neg S$.
  3. If $S_1,S_2,S_3,\dots$ are in $ℳ$, then so is their union ${\bigcup }_i{S}_i$.

  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.

Measurable sets

A measurable space is usually defined to be a set $X$ with a $\sigma$-algebra $ℳ$ on $X$; sometimes one of the more general variations above is used.

In any case, an $ℳ$-measurable subset of $X$, or just a measurable set, is any subset of $X$ that belongs to $ℳ$. If $ℳ$ is one of the more general variations, then we also want some subsidiary notions: $S$ is relatively measurable if $S\cap T$ belongs to $ℳ$ whenever $T$ does, and $S$ is $\sigma$-measurable if it is a countable union of elements of $ℳ$. 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 $ℳ$ 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 $ℳ$ is at least a $\delta$-ring.

Generating $\sigma$-algebras

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 $ℬ$ of sets whatsoever, the intersection of all $\sigma$-algebras containing $ℬ$ is a $\sigma$-algebra, the $\sigma$-algebra generated by $ℬ$. (We can similarly define the $\delta$-ring generated by $ℬ$ and similar concepts for all of the other notions defined above.)

What is missing is a simple description of the $\sigma$-algebra generated by $ℬ$. (For a mere algebra, this is easy; any $ℬ$ can be taken as a subbase of an algebra, the symmetric unions of finite families of elements of $ℬ$ 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:

• Start with a collection ${\Sigma }_0$ (our collection $ℬ$ above), and let ${\Pi }_0$ be the collection of the complements of the members of ${\Sigma }_0$.
• Let ${\Sigma }_1$ be the collection of unions of countably infinite families of sets in ${\Pi }_0$, and let ${\Pi }_1$ be the collection of their complements (the intersections of countably infinite families of sets in ${\Sigma }_0$); even ${\Sigma }_1\cup {\Pi }_1$ is not in general a $\sigma$-algebra.
• Continue by recursively, defining ${\Sigma }_n$ for all natural numbers $n$.
• Let ${\Sigma }_{\omega }$ be the union of the various ${\Sigma }_n$; although this is closed under complement, it is still not in general a $\sigma$-algebra.
• Continue by transfinite? recursion, defining ${\Sigma }_{\alpha }$ for all countable ordinal numbers $\alpha$.
• Let ${\Sigma }_{{\omega }_1}$ be the union of the various ${\Sigma }_{\alpha }$; this is finally a $\sigma$-algebra.

So we need an ${\aleph }_1$ steps, not just $2$.

(This is only the beginning of descriptive set theory; our ${\Sigma }_{\alpha }$ are their ${\Sigma }_{\alpha }^0$ —except that for some reason they start with ${\Sigma }_1^0$ 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.

Examples

• 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 ${ℝ}^N$ for $N$ on the order of Avogadro's number) describing every last detail of a physical system and a macrocanonical phase space $p$ (typically ${ℝ}^2$ or at least ${ℝ}^n$ for $n<10$) 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 $(X,ℳ)$ is equipped with a (positive) measure $\mu$, making it into a measure space, then the sets of measure zero form a $\sigma$-ideal of $ℳ$ (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 $ℳ$ under $\mu$. (Even if $ℳ$ 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 $𝒩$ and simply declaring by fiat that these are the null sets, as with a localisable measurable space; then we speak of the completion of $ℳ$ with respect to $𝒩$ (or sometimes with respect to the $\delta$-filter $ℱ$ 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.

Alternatives

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.