Measure space

Definition

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

• X is a set
• \mathcal A is a σ-algebra on the set X
• \mu is a measure on (X, \mathcal{A})
• \mu must satisfy countable additivity. That is, if (A_{n}){n=1}^{\infty} are pair-wise disjoint then \mu(\cup{n=1}^{\infty}A_{n}) =\sum_{n=1}^{\infty}\mu(A_{n})

In other words, a measure space consists of a measurable space (X, \mathcal{A}) together with a measure on it.

Example

Set X = {0, 1}. The \sigma-algebra on finite sets such as the one above is usually the power set, which is the set of all subsets (of a given set) and is denoted by \wp(\cdot). Sticking with this convention, we set \mathcal{A} = \wp(X)

In this simple case, the power set can be written down explicitly: \wp(X) = {\varnothing, {0}, {1}, {0, 1}}.

As the measure, define \mu by \mu({0}) = \mu({1}) = \frac{1}{2}, so \mu(X) = 1 (by additivity of measures) and \mu(\varnothing) = 0 (by definition of measures).

This leads to the measure space (X, \wp(X), \mu). It is a probability space, since \mu(X) = 1. The measure \mu corresponds to the Bernoulli distribution with p = \frac{1}{2}, which is for example used to model a fair coin flip.

Important classes of measure spaces

Most important classes of measure spaces are defined by the properties of their associated measures. This includes, in order of increasing generality:

• Probability spaces, a measure space where the measure is a probability measure
• Finite measure spaces, where the measure is a finite measure
• \sigma-finite measure spaces, where the measure is a \sigma -finite measure

Another class of measure spaces are the complete measure spaces.

References

References

1. (2008). "Introduction to Empirical Processes and Semiparametric Inference". Springer.
2. (2008). "Probability Theory". Springer.
3. (2008). "Probability Theory". Springer.
4. "Measure space".