σ-algebras

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At the end of the previous section we saw that there was no function $\mathrm{\Xi }$ from $\mathcal{P}(\mathbb{R})$ to $[0,\mathrm{\infty }]$ that is countably additive, translation invariant, and assigns intervals their usual lengths. Beholden to those three properties, we will continue by allowing ourselves to shrink the domain of $\mathrm{\Xi }$ to a specific collection of subsets of $\mathbb{R}$. In preparation for this, we discuss in this section the abstract properties such collections of subsets will need to have.

The definition

A σ-algebra on a set $X$ is any collection of subsets of $X$ satisfying abstract rules. In this course, we will use σ-algebras to keep track of the subsets of $X$ whose size we are permitted to calculate.

Definition

Fix a set $X$. A set $\mathcal{B}\subset \mathcal{P}(X)$ is a σ-algebra if it has all the following properties.

1. $X\in \mathcal{B}$.
2. If $B\in \mathcal{B}$ then $X\setminus B\in \mathcal{B}$.
3. For any sequence $B_1,B_2,B_3,\dots$ of sets in $\mathcal{B}$ the union $$B_1\cup B_2\cup B_3\cup \cdots$$ belongs to $\mathcal{B}$.

For our purposes σ-algebras will serve as the domains of measures. By a measurable space we mean a pair $(X,\mathcal{B})$ where $X$ is a set and $\mathcal{B}$ is a $\sigma$-algebra of subsets of $X$.

Faced with a new collection of axioms defining a mathematical object, we should go through several standard procedures: look at some example, deduce some elementary properties, and describe the morphisms of these objects.

The trivial σ-algebra

For any set $X$ the collection $$\{\varnothing ,X\}$$ is a σ-algebra, called the trivial σ-algebra on $X$. It is trivial in the sense that it doesn't contain any interesting subsets of $X$.

The full σ-algebra

For any set $X$ the collection $$\mathcal{P}(X)$$ is a σ-algebra, called the full σ-algebra on $X$.

The σ-algebra generated by a collection

For any set $X$ and any collection $\mathcal{F}\subset \mathcal{P}(X)$ there is a σ-algebra containing $\mathcal{F}$ that we can think of as the σ-algebra generated by $\mathcal{F}$. It is defined to be $$\sigma (\mathcal{F})=\{A\subset \mathbb{R}:A\mathsf{\text{ in every sigma-algebra that contains }}\mathcal{F}\}$$ and is in fact a σ-algebra.

We will often work with σ-algebras generated by this or that collection of sets. It is useful to think of $\sigma (\mathcal{F})$ as the smallest σ-algebra that contains $\mathcal{F}$. In particular, it is always true that if a σ-algebra $\mathcal{C}$ contains $\mathcal{F}$ then it automatically contains $\sigma (\mathcal{F})$.

The Borel σ-algebra

Amongst the σ-algebras we will work with two stand out for their importance. We introduce the first of these now: the Borel σ-algebra. The second, the Lebesgue σ-algebra on the real line, will be introduced later.

Definition

When $X$ is a topological space we can equip $X$ with a distinguished σ-algebra called the Borel σ-algebra. It is denoted $\mathsf{Bor}(X)$ and defined as the σ-algebra generated by the collection of all open subset of $X$. The subsets of $X$ that belong to $\mathsf{Bor}(X)$ are called Borel subsets of $X$.

We have some experience of what open sets look like from metric spaces and from topology. For one example, every open subset of $\mathbb{R}$ is a countable union of disjoint open intervals. For another, if a point $x$ belongs to a metrically open set then so do all of its nearby friends.

What sort of intuition can we develop for the Borel subsets of a topological space? For example, what feelings can we have for a Borel subset of the real line?

The Borel σ-algebra is generated by the open sets. It therefore contains the closed sets, all unions of countably many closed sets, all intersections of countably many open sets, and so on. It is not feasible to give a concrete description of an arbitrary Borel set. However, as a rule of thumb, a set that can be written by hand will generally be a Borel set.

Example

The set $$G=\{x\in \mathbb{R}:\underset{N\to \mathrm{\infty }}{lim}{\displaystyle \frac{1}{N}}\sum _{n=1}^N{2}^{n}x\mathsf{\text{ mod }}1={\displaystyle \frac{1}{2}}\}$$ is a Borel subset of $\mathbb{R}$. This can be seen using unions and intersections to represent quantifiers in the definition of a limit. We have $$G=\bigcap _{r\in \mathbb{N}}\bigcup _{M\in \mathbb{N}}\bigcap _{M\ge N}\{x\in \mathbb{R}:|{\displaystyle \frac{1}{N}}\sum _{n=1}^N{2}^{n}x\mathsf{\text{ mod }}1-{\displaystyle \frac{1}{2}}|<{\displaystyle \frac{1}{r}}\}$$ which is a countable intersection of a countable union of a countable intersection of open sets.

There are exceptions to the above rule of thumb: the set of all real numbers $x$ for which there is an increasing sequence $i(n)$ in $\mathbb{N}$ such that the continued fraction digist $a(1),a(2),\dots$ of $x$ satisfy that $a(i(n))$ divides $a(i(n+1))$ for all $n\in \mathbb{N}$ is not a Borel set! This example, due to Luzin, is somewhat beyond the material of this course. See this post for more discussion.

We defined the Borel σ-algebra as the σ-algebra generated by the open sets. In fact, it is also the σ-algebra generated by many other families of sets.

Proposition

The σ-algebra on $\mathbb{R}$ generated by the open intervals is the Borel σ-algebra on $\mathbb{R}$.

Proof:

Write $\mathcal{C}$ for the σ-algebra on $\mathbb{R}$ generated by the open intervals. We must prove $\mathcal{C}=\mathsf{Bor}(\mathbb{R})$. Since every open interval is an open subset of $\mathbb{R}$, the Borel σ-algebra contains ever open interval. As $\mathcal{C}$ is the smallest σ-algebra containing the open intervals, it must be smaller than $\mathsf{Bor}(\mathbb{R})$ so $\mathcal{C}\subset \mathsf{Bor}(\mathbb{R})$.

For the reverse inclusion, we need to prove that every Borel set belongs to $\mathcal{C}$. Since every open subset of $\mathbb{R}$ is a countable union of open intervals, every open subset of $\mathbb{R}$ belongs to $\mathcal{C}$. We have therefore shown $\mathcal{C}$ is a σ-algebra that contains every open set. It must then be larger than the smallest such σ-algebra, which is $\mathsf{Bor}(\mathbb{R})$. Thus $\mathcal{C}\supset \mathsf{Bor}(\mathbb{R})$.▮

First properties

To work with and manipulate σ-algebras we will need to absorb several immediate consequences of the axioms.

Lemma

Fix a σ-algebra $\mathcal{B}$.

1. If $B_1,B_2,\dots \in \mathcal{B}$ then $$\bigcap _{n=1}^{\mathrm{\infty }}{B}_n$$ belongs to $\mathcal{B}$.
2. If $B_1,\dots ,B_n\in \mathcal{B}$ then $B_1\cup \cdots \cup B_n\in \mathcal{B}$.
3. If $B_1,\dots ,B_n\in \mathcal{B}$ then $B_1\cap \cdots \cap B_n\in \mathcal{B}$.
4. If $B,C\in \mathcal{B}$ then $B\setminus C\in \mathcal{B}$.

Proof:

Fix a σ-algebra $\mathcal{B}$.

1. This is a consequence of the set-theoretic identity $$\bigcap _{n=1}^{\mathrm{\infty }}{B}_n=X\setminus (\bigcup _{n=1}^{\mathrm{\infty }}X\setminus B_n)$$ and the defining properties of a σ-algebra.
2. Taking $B_{n+i}=\varnothing$ for all $i\in \mathbb{N}$ and applying the countable union property gives the result.
3. Taking $B_{n+i}=X$ for all $i\in \mathbb{N}$ and applying the above countable intersection property gives the result.
4. Write $B\setminus C=B\cap (X\setminus C)$ and apply the complement and finite intersections properties.▮