The Unapologetic Mathematician

Mathematics for the interested outsider

Algebras of Sets

Okay, now that root systems are behind us, I’m going to pick back up with some analysis. Specifically, more measure theory. But it’s not going to look like the real analysis we’ve done before until we get some abstract basics down.

We take some set X, which we want to ultimately consider as a sort of space so that we can measure parts of it. We’ve seen before that the power set P(X) — the set of all the subsets of X — is an orthocomplemented lattice. That is, we can take meets (intersections) U\cap V, joins (unions) U\cup V and complements U^c=X\setminus U of subsets of X, and these satisfy all the usual relations. More generally, we can use these operations to construct differences U\setminus V=U\cap V^c.

Now, an algebra \mathcal{A} of subsets of X will be just a sublattice of P(X) which contains both the bottom and top of P(X): the empty subset \emptyset and the whole set X. The usual definition is that if it contains U and V, then it contains both the union U\cup V and the difference U\setminus V, along with \emptyset and X. But from this we can get complements — U^c=X\setminus U — and DeMorgan’s laws give us intersections — U\cap V=(U^c\cup V^c)^c.

It’s important to note here that these operations let us define finite unions and intersections, just by iteration. But finite operations like this are just algebra. What makes analysis analysis is limits. And so we want to add an “infinite” operation.

Let’s say we have a countably infinite collection of subsets, \{U_i\}_{i=1}^\infty. Then we define the countable union as a limit

\displaystyle\bigcup\limits_{i=1}^\infty U_i=\lim\limits_{n\to\infty}\bigcup\limits_{i=1}^nU_i

We could also just say that the countable union consists of all points in any of the U_i, but it will be useful to explicitly think of this as a process: Starting with U_1 we add in U_2, then U_3, and so on. If x\in U_k for some k, then by the time we reach the kth step we’ve folded x into the growing union. The countable union is the limit of this process.

This viewpoint also brings us into contact with the category-theoretic notion of a colimit (feel free to ignore this if you’re category-phobic). Indeed, if we define V_0=\emptyset and

\displaystyle V_n=\bigcup\limits_{i=1}^nU_i

then clearly we have an inclusion mapping V_i\to V_{i+1} for every natural number i. That is, we have a functor from the natural numbers \mathbb{N} as an order category to the power set P(X) considered as one. And the colimit of this functor is the countable union.

So, let’s say we have an algebra \mathcal{A} of subsets of X and add the assumption that \mathcal{A} is closed under such countable unions. In this case, we say that \mathcal{A} is a “\sigma-algebra”. We can extend DeMorgan’s laws to show that a \sigma-algebra \mathcal{A} will be closed under countable intersections as well as countable unions.

March 15, 2010 Posted by | Analysis, Measure Theory | 9 Comments