# Measure Theory

*Introduction to measure theory*

Measure theory is the study of **measures**, aiming to formalize the *notion of size*.

A measure is defined as a function that assigns a non-negative real number to subsets of a set $X$. Measures must be *countably additive*, meaning the measure of a set that can be decomposed into a finite or countably infinite number of ‘smaller’ subsets must be equal to the sum of the measures on each of the smaller subsets. Formally, a measure is a function $\mu$ mapping from a $\sigma$-algebra $\Sigma$ over a set $X$ to the extended real number line satisfying the following three properties:

**Non-negativity**:

$\forall e \in \Sigma, \mu(e) \ge 0$

**Null empty set**:

$\mu(\emptyset) = 0$

**Countable additivity**:

$\mu\bigg(\bigcup_{i=1}^{\infty}{E_i}\bigg) = \sum_{i=1}^{\infty}{\mu(E_i)}$

Note the axiom of countable additivity is also an axiom $\theta$ of probability theory. In order to assign a consistent measure to all possible subsets of a set, it is typically the case that only trivial measures (like the counting measure) satisfy the necessary properties. This issue is resolved by only defining a measure on a subcollection of all possible subsets; the subsets called **measurable subsets**, which necessarily form a $\sigma$-algebra.

# Measurable Space

The pair $(X, \Sigma)$ is a called a **measurable space**, and the elements of $\Sigma$ are known as **measurable sets**. Note that, unlike a measure space, no measure need be defined (in accordance with its name); it is simply measurable (thanks to $\Sigma$) meaning a measure *can* be defined on the space, but refrains from defining one explicitly.

# Measure Space

A measure space is a triple $(X, \mathcal{A}, \mu)$ where

- $X$ is a non-empty set
- $\mathcal{A}$ is a $\sigma$-algebra on $X$
- $\mu$ is a measure defined on the measurable space $(X, \mathcal{A})$

A measure space includes the necessary components (a measure and a measurable space) for consistently defining a measure in an appropriate context.

# $\sigma$-algebra

Formally, a $\sigma$-algebra on a set $X$ is a collection $\Sigma$ of subsets of $X$ (including $X$ itself) that are closed under complement and countable unions. That is, the complement of any subset in $\Sigma$ is also in $\Sigma$, and the union of any countable number of subsets in $\Sigma$ is also in $\Sigma$. A $\sigma$-algebra is a type of algebra of sets, and is useful in measure theory as the collection of subsets for which a measure is defined (*measureable sets*) is necessarily a $\sigma$-algebra.

## Example

For the set $X = \{a,b,c,d\}$, an example of a $\sigma$-algebra on $X$ is

$\Sigma = \{ \emptyset, \{a,b\}, \{c,d\}, \{a,b,c,d\}\}$

Notice that any element of $\Sigma$ has its complement in $\Sigma$, and any countable union of the elements of $\Sigma$ is also in $\Sigma$.