What is complementation wrt a sigma-algebra?

[Rasalhague]

What is "complementation" wrt a sigma-algebra?

By definition, a $\sigma$-algebra over a set X is a nonempty collection $\Sigma$ of subsets of X (including X itself) that is closed under complementation and countable unions of its members.
http://en.wikipedia.org/wiki/Σ-algebra

What does complementation mean here? Is the statement saying that the complement of a subset of X in $\Sigma$ must also be in $\Sigma$ for $\Sigma$ to qualify as the underlying set of a $\sigma$-algebra?

$$A \in \Sigma \Rightarrow A \subset X$$

and

$$A \in \Sigma \Rightarrow \enspace \{ x : x \in X, x \notin A \} \in \Sigma$$

And is "collection" just a convenient synonym for set?

 

[CompuChip]

Yep. "Complementation" means taking the complement of a set in the algebra, "closed under some operation" means that the operation applied to any element gives another element (just like groups are closed under the group operation and vector spaces are closed under addition).

Usually the word "collection" is used for a set whose elements are themselves sets. For example, "the collection of all subsets of R".

 

[Rasalhague]

Thanks CompuChip! Oh, another question: what exactly does algebra mean in this context. Is this "an algebra, F, over a set, X", defined as (X,S), where S is "a non-empty subset of the power set of X closed under the intersection and union of pairs of sets and under complements of individual sets" ( http://en.wikipedia.org/wiki/Algebra_over_a_set ). And is this, in some way, a special case of Mathworld's definition of an algebra as "a vector space [...] with a multiplication" ( http://mathworld.wolfram.com/Algebra.html ), or Wikipedia's definition of an algebra over a field as "a vector space equipped with a bilinear vector product" ( http://en.wikipedia.org/wiki/Algebra_over_a_field )? Or are "algebra over a set" and "algebra over a field" different things?

Wikipededia also gives "sigma-field" and "Borel-field" as synomyms for sigma-algebra. Is it a field in the sense that the real numbers with the standard addition and multiplication are a field? If so what are its addition and multiplication?

 

[mathman]

An algebra in set theory is a collection closed under complementation and FINITE unions and intersections. This is completely different from the other definition of an algebra.

The term field is used sometimes by making union the analog of addition and intersection the analog of multiplication. Don't take it seriously.

 

[Rasalhague]

Thanks, Mathman. I suppose one way in which the analogy breaks down is the fact that complementation, in this sense (absolute complement), isn't a binary operation of the form C:SxS-->S.

 

[CompuChip]

It doesn't really matter, does it?

Formally:
Consider a "universe" X. Let S be a subset of X and f: Xⁿ -> X an (n-ary) operation on X, where
$$X^n := \underbrace{X \times X \times \cdots \times X}_{n\text{ times}}$$

We say that S is closed with respect to f, if
$$f(S^n) \subseteq S$$,
i.e.
$$f(s_1, s_2, \ldots, s_n) \in S$$
for any s_i (i = 1, 2, .., n) in S.

 

[Landau]

An algebra in set theory is a collection closed under complementation and FINITE unions and intersections. This is completely different from the other definition of an algebra.

The term field is used sometimes by making union the analog of addition and intersection the analog of multiplication. Don't take it seriously.

A ring of sets is a ring in the algebraic sense by taking multiplication to be intersection, and + to be symmetric difference.

 

[Rasalhague]

It doesn't really matter, does it?

It being any or all of my questions? Well, in the grand scheme I don't know, but they mattered to me enough to ask. Luke Skywalker voice: "I care!" Learning a new subject often throws up a lot of jargon. It's easy to get lost in it. I find it helps me get my bearings to know what some of it means ;-) Once you already know the subject, you can look at an introductory text and think, well, here's an irrelevant sideline or something trivial, no need to worry about that. But when you're just starting out, you have to sift through all these new names and information, and you don't necessarily know what's important. Noticing familiar names like "algebra" and "field", it seemed natural to ask whether these terms match up to how they're used in other areas I've studied. If so, that would make a pattern that would make it easier to learn. And since they're not (if they're not), I'm glad I did and grateful for all your answers, as they've saved me reading on under the misapprehansion that a sigma algrabra/field might be an algebra in the sense of a "vector space with a multiplication" or a field in the sense of a commutative ring with multiplicative inverses. Even if it doesn't actually make a big difference to using these structures in practice, knowing this means that I can concentrate on learning about what kind of entities they are without being distracted with wondering what their names are meant to suggest.

Or was it just my reference to complementation, in this context, not being a binary operation that you were saying "doesn't matter" in the sense that it has no bearing on whether it could qualify as either of the operations of a field. I thought both had to be binary.

 

[Rasalhague]

A ring of sets is a ring in the algebraic sense by taking multiplication to be intersection, and + to be symmetric difference.

Interesting. And intersection is commutative, so that just leaves the requirement for a multiplicative inverse for all elements of S before such a structure would qualify as a field in the "monoid-group-ring..." sense of the word:

$$\forall A \in S, \, \exists B \in S : A \cap B = E_{\cap}$$

where $E_{\cap}$ is the identity intersection, except when A is the additive identity. I'll have a think about this. Are you saying, though, that the term "field of sets" (deceptively) doesn't imply this? (I.e. that a elements of the underlying set/collection of a "field of sets" don't necessarily have an intersective inverse, whatever that would be.)

 

[Rasalhague]

http://en.wikipedia.org/wiki/Sigma_ring

This article defines a collection of a sets as a sigma ring if it's closed under union and relative complementation, so I was I was wrong in my first post to guess that the sigma algebra article meant absolute complementation?

 

[Rasalhague]

Ring of sets. (A ring.)
Addition: symmetric difference.
Multiplication: intersection.

This definition from Wikipedia ( http://en.wikipedia.org/wiki/Ring_of_sets ).

(1) (a) Isn't it possible to have a ring, elements of whose underlying set are sets, but whose addition and multiplication are some other operation; (b) would you call such an entity a "ring of sets" too, (c) or if not, what? (d) Or is there a theorem that a ring, elements of whose underlying set are sets, is a ring of sets by the above definition if it's a ring, in the "group, ring, field..." sense with any other operations?

(2) The articles on sigma ring and delta ring, linked to from Ring of sets each say that relaxing some condition for the structure in question results in a structure that is a ring, and the word "ring" in each of them links back to Ring of sets, which suggests that a sigma ring and a delta ring are both rings in this sense, and, if so, also rings in the "group, ring, field..." sense. Are they?

(3) (a) Are relative complementation (set-theoretic difference) and countable union thought of literally as the addition and multiplication of sigma rings and delta rings, (b) and if so, which is which: union for addition and intersection for multiplication, as in #3 re. a sigma field? (c) I suppose if closure under these operations implies that a structure is a ring with symmetric difference as addition and intersection as multiplication, there'd be no need to assign those other operations (rel. comp. and countable union) to the same roles. Are sigma rings and delta rings "rings twice over", with two pairs of operations?