I am following the Measure Theory series by D.H. Fremlin and blogging my notes here.

Borel Sets

To understand the definition of Borel Sets, we need to understand two things first:

Generating a $\sigma$-algebra

In order to understand what generating a $\sigma$-algebra means, we will start with a proof.

Let $\mathfrak S$ be a family of $\sigma$-algebras of subsets of a set $X$, i.e.,

$\mathfrak S=\{\Sigma:\Sigma \text{ is a }\sigma\text{-algebra of subsets of }X\}.$

So $\mathfrak S$ is basically a set of set of sets i.e. $\mathfrak S \subseteq \mathcal P(\mathcal PX)$. Then $\bigcap\mathfrak S$ is the intersection of all the $\sigma$-algebras in $\mathfrak S$. We want to prove that $\bigcap\mathfrak S$ is also a $\sigma$-algebra.

Proof.

1. $\emptyset\in\Sigma$ for every $\Sigma\in\mathfrak S$, so $\emptyset\in\bigcap\mathfrak S$.
2. If $E\in\bigcap\mathfrak S$ then $E\in\Sigma$ for every $\Sigma\in\mathfrak S$, so $X\setminus E\in\Sigma$ for every $\Sigma\in\mathfrak S$ and $X\setminus E\in\bigcap\mathfrak S$.
3. Let $\langle E_n\rangle_{n\in\Bbb N}$ be any sequence in $\bigcap\mathfrak S$. Then for every $\Sigma\in\mathfrak S$, $\langle E_n\rangle_{n\in\Bbb N}$ is a sequence in $\Sigma$, so $\bigcup_{n\in\Bbb N}E_n\in\Sigma$; as $\Sigma$ is arbitrary, $\bigcup_{n\in\Bbb N}E_n\in\bigcap\mathfrak S$.

Now we can understand what generating a $\sigma$-algebra means: The $\sigma$-algebra generated by a set $\mathcal A$ is the smallest possible $\sigma$-algebra that contains $\mathcal A$. More precisely,

Let $\mathcal A$ be any family of subsets of $X$. Consider

$\mathfrak S=\{\Sigma:\Sigma \text{ is a }\sigma\text{-algebra of subsets of }X \text{ and } \mathcal A\subseteq\Sigma\}$

Then, the $\sigma$-algebra of subsets of $X$ generated by $\mathcal A$ is $\Sigma_{\mathcal A} = \bigcap\mathfrak S$. Let that sink in.

Another way of obtaining $\Sigma_{\mathcal A}$ from $\mathcal A$ could be: We start off with an empty set, say, $\Sigma_{\mathcal A’}$. We first add $\emptyset$ and every element of $\mathcal A$ to $\Sigma_{\mathcal A’}$. Then we add the complement of every element in $\Sigma_{\mathcal A’}$ to itself. Finally, we add the union of all sequences in $\Sigma_{\mathcal A’}$ to itself. Then the set $\Sigma_{\mathcal A’}$ is actually $\Sigma_{\mathcal A}$.

Examples

• For any $X$, the $\sigma$-algebra of subsets of $X$ generated by $\emptyset$ is $\{\emptyset,X\}$.
• The $\sigma$-algebra of subsets of $\Bbb N$ generated by $\{\{n\}:n\in\Bbb N\}$ is $\mathcal P\Bbb N$.

Open Sets

A set $S \subseteq \Bbb R$ is considered open if $\forall\,\, s\in S \,\,\exists\,\, \delta > 0$ such that $(s-\delta, s+\delta)\in S$.

Here’s a more general definition (with regards to the dimension of the Euclidean space): A set $S \subseteq \Bbb R^r;\,\, r \in \Bbb Z^+$ is considered open if $\forall\,\, s\in S \,\,\exists\,\, \delta > 0$ such that $\{t: ||(s-t)|| < \delta\}\subseteq S$ where $||(s-t)||$ denotes the Euclidean distance between $s$ and $t$ (i.e. all points within a distance $\delta$ from $s$ lie in $S$).

Borel Sets

The Borel sets of $\Bbb R$, are just the members of the $\sigma$-algebra of subsets of $\Bbb R$ generated by the family of open sets of $\Bbb R$; the $\sigma$-algebra itself is called the Borel $\sigma$-algebra.

In other words, we pick all the open sets in $\mathcal P\Bbb R$ and put them into a set, say, $\mathcal A$. Then using $\mathcal A$, we generate a $\sigma$-algebra $\Sigma_{\mathcal A}$ of the subsets of $\Bbb R$. The members of $\Sigma_{\mathcal A}$ are called Borel sets and $\Sigma_{\mathcal A}$ is called the Borel $\sigma$-algebra.

For a more general definition (with regards to the dimension of the Euclidean space), replace $\Bbb R$ with $\Bbb R^r;\,\,r\in\Bbb Z^+$ in the above two paragraphs.