σ-Algebra

If we want to have some function \(\mu\) on subsets of a set \(X\), such that \(\mu\) satisfies some properties we expect of a function that "measures" "sizes" of the subsets, it turns out that the power set of \(X\) is not satisfactory - we need a set of subsets with more structure, namely, a σ-algebra.

A σ-algebra \(\Sigma\) on some set \(X\) is a set of subsets where:

• \(\emptyset \in \Sigma\) (or equivalently \(X \in \Sigma)\)
• \(A \in \Sigma \implies X\textbackslash A \in \Sigma\)
• \(A_1,A_2,\cdots \in \Sigma \implies \bigcup_i A_i \in \Sigma\)

A σ-algebra over \(X\) could also be defined as an algebra over \(X\) where the finite union is replaced by the countable union.

The elements (i.e. sets) in the σ-algebra are called measurable sets.

In practice, to construct a σ-algebra one selects a collection of subsets \(E \subset \mathcal{P}(X)\) and then defines the σ-algebra of interest to be \(\sigma(E) := \bigcap_{\Sigma_i : \Sigma_i \text{ is a } \sigma-\text{algebra and } E \subseteq \Sigma_i}\Sigma_i\), which is also called the σ-algebra generated by \(E\), which is in turn a σ-algebra because intersections of σ-algebras are also σ-algebras. In fact, all σ-algebras can be written as \(\sigma(E)\) for some \(E\).

The pair \((X,\Sigma)\) is called a measurable space, and when equipped with a measure \(\mu\), the triple \((X,\Sigma,\mu)\) is called a measure space, if \(\mu(X)<\infty\) then \((X,\Sigma,\frac{\mu}{\mu(X)}: \Sigma \to [0,1])\) is a probability space.