nLab
sigma-algebra

σ\sigma-Algebras

Idea

σ\sigma-algebras and their variants are collections of subsets important in classical measure theory and probability theory.

Although σ\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 XX, it is often useful to consider additional (typically coarser) σ\sigma-algebras of measurable subsets of XX.

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 XX, a σ\sigma-algebra is a collection of subsets of XX that is closed under complementation and under unions and intersections of countable families.

Notice that the power set 𝒫X\mathcal{P} X of XX 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 σ\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 XX and a collection \mathcal{M} of subsets SXS \subseteq X, there are really several kinds of collections that \mathcal{M} could be:

  • A ring on XX is a collection \mathcal{M} which is closed under relative complementation and under unions of finitary families. That is:

    1. The empty set \empty is in \mathcal{M}.
    2. If SS and TT are in \mathcal{M}, then so is their union STS \cup T.
    3. If SS and TT are in \mathcal{M}, then so is their relative complement TST \setminus S.

    It follows that \mathcal{M} is closed under intersections of inhabited finite families and under symmetric difference of finite families: * If SS and TT are in \mathcal{M}, then so is their intersection ST=T(TS)S \cap T = T \setminus (T \setminus S). * If SS and TT are in \mathcal{M}, then so is their symmetric difference ST=(TS)(ST)S \uplus T = (T \setminus S) \cup (S \setminus T).

    We can actually use the latter as an alternative to (2), since ST=(ST)(ST)S \cup T = (S \uplus T) \uplus (S \cap T). Or we can use the pair as an alternative to (2,3), since TS=(ST)TT \setminus S = (S \cap T) \uplus T. For that matter, we can weaken (1) to simply say that some set SS is in \mathcal{M}; then =SS\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 SS 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 XX is simply a subring (in this sense) of the Boolean ring 𝒫X\mathcal{P} X.

  • A δ\delta-ring on XX is a ring (as above) \mathcal{M} which is closed under intersections of countably infinite families. That is:

    1. The empty set \empty is in \mathcal{M}.
    2. If SS and TT are in \mathcal{M}, then so is their union STS \cup T.
    3. If SS and TT are in \mathcal{M}, then so is their relative complement TST \setminus S.
    4. If S 1,S 2,S 3,S_1, S_2, S_3, \ldots are in \mathcal{M}, then so is their intersection iS i\bigcap_i S_i.

    Of course, every δ\delta-ring is a ring, but not conversely. Actually, if you want to define the concept of δ\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 ST=STTTS \cap T = S \cap T \cap T \cap T \cap \cdots).

    The symbol ‘δ\delta’ here is from German ‘Durchschnitt’, meaning intersection; it may be used in many contexts to refer to intersections of countable families.

  • A σ\sigma-ring on XX is a ring (as above) \mathcal{M} which is closed under unions of countably infinite families. That is:

    1. The empty set \empty is in \mathcal{M}.
    2. If SS and TT are in \mathcal{M}, then so is their union STS \cup T.
    3. If SS and TT are in \mathcal{M}, then so is their relative complement TST \setminus S.
    4. If S 1,S 2,S 3,S_1, S_2, S_3, \ldots are in \mathcal{M}, then so is their union iS i\bigcup_i S_i.

    Now (2) is simply redundant; ST=STTTS \cup T = S \cup T \cup T \cup T \cup \cdots. A σ\sigma-ring is obviously a ring, but in fact it is also a δ\delta-ring; iS i=( iS i) j( iS iS j)\bigcap_i S_i = (\bigcup_i S_i) \setminus \bigcup_j (\bigcup_i S_i \setminus S_j).

    The symbol ‘σ\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 XX is a ring (as above) \mathcal{M} to which XX itself belongs. That is:

    1. The empty set \empty is in \mathcal{M}.
    2. If SS and TT are in \mathcal{M}, then so is their union STS \cup T.
    3. If SS and TT are in \mathcal{M}, then so is their relative complement TST \setminus S.
    4. The improper subset XX is in \mathcal{M}.

    Actually, (2) is now redundant again; ST=X((XT)S)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 XX); that is:

    • If SS is in \mathcal{M}, then so is its complement ¬S\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 TS=¬(S¬T)T \setminus S = \neg(S \cup \neg{T}) and (4) follows because X=¬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 {,X}\{\empty, X\} (for an inhabited set XX).

  • Finally, a σ\sigma-algebra or σ\sigma-field on XX is a ring \mathcal{M} that is both an algebra (as above) and a σ\sigma-ring (as above). That is:

    1. The empty set \empty is in \mathcal{M}.
    2. If SS and TT are in \mathcal{M}, then so is their union STS \cup T.
    3. If SS and TT are in \mathcal{M}, then so is their relative complement TST \setminus S.
    4. The improper subset XX is in \mathcal{M}.
    5. If S 1,S 2,S 3,S_1, S_2, S_3, \ldots are in \mathcal{M}, then so is their union iS i\bigcup_i S_i.

    As with σ\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 σ\sigma-algebra states: 1. The empty set \empty is in \mathcal{M}. 2. If SS is in \mathcal{M}, then so is its complement ¬S\neg{S}. 3. If S 1,S 2,S 3,S_1, S_2, S_3, \ldots are in \mathcal{M}, then so is their union iS i\bigcup_i S_i.

    And again we could again just as easily use intersection as union, even in the infinitary axiom. That is, a δ\delta-algebra is automatically a σ\sigma-algebra, because iS i=¬ i¬S i\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 σ\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, δ\delta‑ or σ\sigma-rings may be more convenient than σ\sigma-algebras for some purposes; for example, vector-valued measures on δ\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 δ\delta-ring, while the collection of measurable sets with σ\sigma-finite measure is a σ\sigma-ring.

Measurable sets

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

In any case, an \mathcal{M}-measurable subset of XX, or just a measurable set, is any subset of XX that belongs to \mathcal{M}. If \mathcal{M} is one of the more general variations, then we also want some subsidiary notions: SS is relatively measurable if STS \cap T belongs to \mathcal{M} whenever TT does, and SS is σ\sigma-measurable if it is a countable union of elements of \mathcal{M}. Notice that every relatively measurable set is measurable iff SS is at least an algebra; in any case, the relatively measurable sets form a (σ\sigma)-algebra if \mathcal{M} is at least a (δ\delta)-ring. Similary, every σ\sigma-measurable set is measurable iff SS is at least a σ\sigma-ring; in any case, the σ\sigma-measurable sets form a σ\sigma-ring if \mathcal{M} is at least a δ\delta-ring.

Generating σ\sigma-algebras

As a σ\sigma-algebra is a collection of subsets, we might hope to develop a theory of bases and subbases of σ\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 δ\delta-rings and σ\sigma-rings is just as tricky as generating σ\sigma-algebras.)

We do get something by general abstract nonsense, of course. It's easy to see that the intersection of any collection of σ\sigma-algebras is itself a σ\sigma-algebra; that is, we have a Moore closure. So given any collection \mathcal{B} of sets whatsoever, the intersection of all σ\sigma-algebras containing \mathcal{B} is a σ\sigma-algebra, the σ\sigma-algebra generated by \mathcal{B}. (We can similarly define the δ\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 σ\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 δ\delta-ring to a σ\sigma-algebra, nothing this simple will work.)

In fact, the question of how to generate a σ\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 Σ 0\Sigma_0 (our collection \mathcal{B} above), and let Π 0\Pi_0 be the collection of the complements of the members of Σ 0\Sigma_0.
  • Let Σ 1\Sigma_1 be the collection of unions of countably infinite families of sets in Π 0\Pi_0, and let Π 1\Pi_1 be the collection of their complements (the intersections of countably infinite families of sets in Σ 0\Sigma_0); even Σ 1Π 1\Sigma_1 \cup \Pi_1 is not in general a σ\sigma-algebra.
  • Continue by recursively, defining Σ n\Sigma_n for all natural numbers nn.
  • Let Σ ω\Sigma_\omega be the union of the various Σ n\Sigma_n; although this is closed under complement, it is still not in general a σ\sigma-algebra.
  • Continue by transfinite? recursion, defining Σ α\Sigma_\alpha for all countable ordinal numbers α\alpha.
  • Let Σ ω 1\Sigma_{\omega_1} be the union of the various Σ α\Sigma_\alpha; this is finally a σ\sigma-algebra.

So we need an 1\aleph_1 steps, not just 22.

(This is only the beginning of descriptive set theory; our Σ α\Sigma_\alpha are their Σ α 0\Sigma^0_\alpha —except that for some reason they start with Σ 1 0\Sigma^0_1 instead of Σ 0 0\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 XX is closed under all operations, so it is a σ\sigma-algebra.

  • If XX is a topological space, the σ\sigma-algebra generated by the open sets (or equivalently, by the closed sets) in XX is the Borel σ\sigma-algebra; its elements are called the Borel sets of XX. 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 PP (typically something like N\mathbb{R}^N for NN on the order of Avogadro's number) describing every last detail of a physical system and a macrocanonical phase space pp (typically 2\mathbb{R}^2 or at least n\mathbb{R}^n for n<10n \lt 10) describing those features of the system that can be measured practically, with a projection PpP \to p. Then the preimage under this projection of the Borel σ\sigma-algebra of pp is a σ\sigma-algebra on PP, and the thermodynamic entropy of the system is (theoretically) its information-theoretic entropy with respect to this σ\sigma-algebra.

  • If a measurable space (X,)(X,\mathcal{M}) is equipped with a (positive) measure μ\mu, making it into a measure space, then the sets of measure zero form a σ\sigma-ideal of \mathcal{M} (that is an ideal that is also a sub-σ\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 σ\sigma-ideal in the power set of XX. Call a set μ\mu-measurable if it is the union of a measurable set and a null set; then the μ\mu-measurable sets form a σ\sigma-algebra called the completion of \mathcal{M} under μ\mu. (Even if \mathcal{M} is only a δ\delta-ring, still the null sets will form a σ\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 σ\sigma-ideal 𝒩\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 𝒩\mathcal{N} (or sometimes with respect to the δ\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 σ\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 σ\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 σ\sigma-algebras of measurable sets in a measurable space. One solution may to use quotient measurable spaces in place of sub-σ\sigma-algebras; for example, see explicit quotient in the example of macroscopic entropy above.

Revised on February 18, 2013 16:31:56 by Anonymous Idiot? (89.17.128.124)