An attempt to codify what "reasonable" sets can be constructed. A large chunk of mathematics cannot be done merely in a set algebra (which see). For instance, the set algebra generated by all open balls in Rn (i.e. the smallest set algebra which has as elements each open ball) is quite complicated: it contains all finite unions of "lenses" (my term), where each "lens" is a finite intersection of open balls. Not only are these sets complicated to describe, they're not interesting for almost any problem. And if we used slightly different objects (say open ellipsoids, or closed balls), we'd get a very different algebra.

If, on the other hand, we allow ourselves to take countable unions and intersections, we get a much nicer set. First, every open set in Rn is a countable union of open balls, so we'll have all open sets. It follows that we could have started from any base for the open sets in Rn to get this algebra.

The algebra (known as the algebra of Borel sets) contains essentially all sets you can explicitly construct. It's also the basis for measure theory, probability, descriptive set theory, and any number of other important branches of Mathematics.

So let's describe what a σ-algebra is (and before that, I should note that the squiggle "σ" is a lowercase sigma Σ).

Definition

Let X be a set. A σ algebra on X is a set B of subsets of X such that:
  • X ∈ B
  • If A ∈ B then also its complement X\A ∈ B (B is closed under complement)
  • If A1,A2,... ∈ B and A is their union then also A ∈ B. (B is closed under countable union)
By de Morgan's laws, B is also closed under countable intersection.

2X, the set of all subsets of X, is always a σ algebra -- it's closed under any union. This is the "largest" σ algebra. {Ø,X} is also a σ algebra -- it's the "smallest" one.

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