# 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.