The main use of $\sigma$-algebras are in the definition of measures.

Definition 0: Power Set

The power set $P(A)$ of set $A$ is the set of all possible subsets and the empty set. For example, if $A=\{1,2,3\}$ then $P(A)=\{ \{\emptyset\},\{1,2,3\},\{1,2\},\{2,3\},\{1,3\},\{1\},\{2\},\{3\}\}$.

One can verify that $|P(A)| = 2^{|A|}$.

Definition 1: $\sigma$-Algebras

A collection $A$ of subsets of a set $X$ is a $\sigma$-Algebra iff:

  1. $\emptyset,A \in A$
  2. $B \in A \Rightarrow B^C \in A$
  3. $A_i \in A , i \in \mathbb{N} \in \Rightarrow \bigcup_{i=0}^{\infty} A_i \in A$

Any set $A_i \in A$ is known as a $A$-measurable set.

Remark 1.

It follows from the definition above that a countable intersection of sets in $A$ is also in $A$ (DeMorgan’s Law and Rules #3 and #2 above).

Generating $\sigma$-Algebras

For $M \subseteq P(A)$ where $M$ is a subset of the elements in a power set $P(A)$, there exists a smallest $\sigma$-algebra that contains $M$.

$\sigma (M) = \bigcap_{A \supseteq M}A$ where $A$ are $\sigma$-algebras. Thus, according to Remark 1. $\sigma (M)$ is also a $\sigma$-algebra.

In other words $\sigma (M)$ is the $\sigma$-algebra generated by $M$. While we are focused on the smallest $\sigma$-algebra, a trivial $\sigma$-algebra that contains $M$ is simply the power set $P(A)$. However, we want the most efficient $\sigma$-algebra. The $\sigma$-algebra generated by $M$ can be easily found once we know $M$ given the rules of $\sigma$-algebra. Also, simply following these rules will lead us to the most efficient version as we will see below.

Example: Let $A=\{a,b,c,d\}$ and $M=\{\{a\},\{b\}\}$.

$\sigma (M) = \{\emptyset,A,\{a\},\{b\},\{a,b\},\{b,c,d\},\{a,c,d\},\{c,d\}\}$

Where did this come from? The first subsets two come from rule #1 of $\sigma$-algebra. The next three subsets come from the countable union of subsets. The final three sub-sets come from the complement of the first 5 subsets, knowing that $\emptyset^C=A$.

You can go through each of the three rules of $\sigma$-algebra and prove that $\sigma (M)$ is a $\sigma$-algebra.

Open Sets

Open sets are an abstract way of thinking of topological separation. In the most concrete sense, we have the Euclidean metric which can be used to define an open-set and give an exact metric to decide elements that exist within the open set and those that are separate. However, we can continue to abstract the idea of separation such that rather than a concrete metric we could consider separation based on setness alone. For example, if two elements do not exist in the same open set generated about a third element, then there is a degree of separateness between the two elements. To understand this abstraction at a deeper level, we will look at how open sets can be generated in the most concrete form in Euclidean space and then extend it to metric and topological spaces.

Euclidean Space

An open set is an abstract way of thinking about open intervals on the real line (e.g. $I=(0,1)$). In n-dimensional Euclidean space or $\mathbb{R}^n$ an open set $U$ is open if for every n-dimensional point around $x$ there exists a positive real number $\epsilon$ such that the Euclidean distance between $x$ and all points in $U$ is less than $\epsilon$.

In other words, we could image an n-dimensional ball of radius $\epsilon$ centered at $X$ encapsulating all points in $U$.

Metric Space

We can further abstract this by defining some metric $d$ to replace the Euclidean metric remembering that a metric is defined as a function that is non-negative (e.g. $d(x,y)\geq0$), symmetrical (e.g. $d(x,y)=d(y,x)$), and satisifies the triangle inequality (e.g. $d(x,y)+d(y,z) \leq d(x,z)$).

A subset $U$ on a metric space $(M,d)$ is open if for any point $x$ in $U$ there exists a real number $\epsilon >0$ such that any other point in $y \in M$ such that $d(x,y) < \epsilon$ implies $y \in U$.

In other words, given a reference element in the metric space and some distance metric, we can generalize the notion of a neighborhood to define an open set.

Topological Space

The final abstraction is in a topological space in which a topology is defined to generalize the notion of a space in that a topology $T$ is defined as a collection of subsets that are considered open.

If $X$ is a set, a topology $T$ on $X$ is defined as the subsets of $X$. The elements of $T$ are open if

  1. $X,\emptyset \in T$
  2. Any union of the elements (open sets) of $T$ exists in $T$ and is thus also an open set
  3. Any finite intersection of the elements of $T$ exist in $T$ and are also open.

Infinite intersections cannot be guaranteed to produce open set (e.g. the infinite intersection of the intervals $(\frac{-1}{n},\frac{1}{n})$ is the set $\{0\}$ which is not open.

You may notice that a topological space does not include any definition of $d$ but rather focuses entirely on abstraction of open sets. This is because a metric space can be thought of as a topological space consisting of only unions of open balls determined by the metric function $d$.

Now that the idea of a topological space $T$ has been defined and can be used to work backward to a more concrete version like Euclidean space, we can better understand what a Borel $\sigma$-algebra looks like.

Definition 2: Borel $\sigma$-algebra

Simply put, the Borel $\sigma$-algebra $B(A)$ is the $\sigma$-algebra generated by the open sets of $A$ where $A$ is a topological space for example a metric space or even $\mathbb{R}^n$. These open sets are captured by the topological structure $T$ such that

\[B(A) := \sigma (T)\]

In other words, its the most efficient enumeration of the open sets of the topological space that satisfy the three requirements of $\sigma$-algebra. While $B(A)$ is obviously very large, it is more efficient than the power set $P(A)$.