# 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 $$0$$.

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

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

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.