In a metric space, the structure was defining a value for each two elements of the set.
In a measure space, the structure defines a value of subsets of the set.
A measure space includes the set \(X\), subsets of the set, \(\Sigma\), and a function \(\mu\) which maps from \(\Sigma\) to \(\mathbb{R}\).
Requirement for \(\Sigma\).
\(\forall E \in \Sigma : \mu (E)\ge 0\)
\(\mu (\null )=0\)
\(\mu (\lor_{k=1}^{\infty} E_k)=\sum \mu (E_k)\)
Where all elements \(E_k\) are disjoint. That is, they have no elements in common.
\(\mu (E)\)
This provides the number of elements in \(E\).
This is discussed in more detail in Statistics.