Measure Theory

Introduction to measure theory
created: 2019-03-02 · modified: 2021-03-05

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 XX. 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 μ\mu mapping from a σ\sigma-algebra Σ\Sigma over a set XX to the extended real number line satisfying the following three properties:

  1. Non-negativity:

∀e∈Σ,μ(e)≥0\forall e \in \Sigma, \mu(e) \ge 0

  1. Null empty set:

μ(∅)=0\mu(\emptyset) = 0

  1. Countable additivity:

μ(⋃i=1∞Ei)=∑i=1∞μ(Ei)\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 θ\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 σ\sigma-algebra.

Measurable Space

The pair (X,Σ)(X, \Sigma) is a called a measurable space, and the elements of Σ\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 Σ\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,A,μ)(X, \mathcal{A}, \mu) where

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

σ\sigma-algebra

Formally, a σ\sigma-algebra on a set XX is a collection Σ\Sigma of subsets of XX (including XX itself) that are closed under complement and countable unions. That is, the complement of any subset in Σ\Sigma is also in Σ\Sigma, and the union of any countable number of subsets in Σ\Sigma is also in Σ\Sigma. A σ\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 σ\sigma-algebra.

Example

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

Σ={∅,{a,b},{c,d},{a,b,c,d}}\Sigma = \{ \emptyset, \{a,b\}, \{c,d\}, \{a,b,c,d\}\}

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