# The Unapologetic Mathematician

## 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 $k$th 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

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2. “The countable union is the limit of this process.”

Is ‘limit’ referring to some rigorous concept here, or does the rigor come from the characterization in terms of colimits?

Comment by Avery Andrews | March 17, 2010 | Reply

3. It’s sort of a subtle point, Avery. I’d say that without reference to the colimit construction, the “limit” can be construed as a suggestive term from natural language.

Really we can just define the union of an arbitrary family on axiomatic set theory ground. Conceiving of it as a process is a helpful viewpoint.

Comment by John Armstrong | March 17, 2010 | Reply

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