Measure space

Defining measure spaces

Measure space

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 \(\mathbb{R}\).

Sigma algebra

Requirement for \(\Sigma \).

Axioms for measures

Measures are non-negative

\(\forall E \in \Sigma : \mu (E)\ge 0\)

The measure for the null set is \(0\).

\(\mu (\null )=0\)

Disjoint sets are additive

\(\mu (\lor_{k=1}^{\infty} E_k)=\sum \mu (E_k)\)

Where all elements \(E_k\) are disjoint. That is, they have no elements in common.

Examples of measure spaces

The counting measure

\(\mu (E)\)

This provides the number of elements in \(E\).

The probability measure

This is discussed in more detail in Statistics.