Measures#

Definition: Measurable Space

A measurable space \((X, \Sigma)\) is a set \(X\) equipped with a σ-algebra \(\Sigma\).

A measurable space is simply the combination of the entire thing \(X\) which we have and the parts of it that we can measure.

Definition: Measure

Let \((X, \Sigma)\) be a measurable space.

A measure on \((X, \Sigma)\) is a function \(\mu: \Sigma \to \mathbb{R}_{\ge 0} \cup \{\infty\}\) from \(\Sigma\) to the non-negative extended real numbers with the following properties:

\(\mu(\varnothing) = 0\);
If \((S_i)_{i \in \mathbb{N}}\) is a sequence of pairwise disjoint measurable sets, then \(\mu\) of their union is the sum of \(\mu(S_i)\).

\[\mu\left(\bigcup_{i =1}^{\infty} S_i \right) = \sum_{i =1}^{\infty} \mu(S_i)\]