Measure space

A measure space is an important idea used in measure theory, a part of mathematics that looks at ways to find the size or volume of different things. It has three main parts: a basic set of items, certain groups of these items that can be measured (called a σ-algebra), and a way to actually measure them (called the measure).

One common example of a measure space is a probability space, which helps us understand chances and likelihoods in games, science, and many other areas.

A related idea is called a measurable space, which has the first two parts—a set and its measurable groups—but does not include a specific way to measure them. This helps mathematicians study the basics before adding the measuring part.

Definition

A measure space is a special way to study sizes or amounts in math. It has three parts:

• A set: This is just a collection of things we want to measure.
• A list of special groups: These are groups of items from the set that we can measure.
• A measuring tool: This tells us the size or amount for each special group.

One important example of a measure space is a probability space, which helps us understand chances and likelihoods.

Example

In this example, we look at a simple set with just two items: 0 and 1. We want to measure these items, so we consider all possible groups of these items. These groups include having nothing, just 0, just 1, or both 0 and 1 together.

We then define a way to measure these groups. For the groups containing just one item (either 0 or 1), we assign a value of one-half. The group with both items gets a measure of 1, and the empty group gets a measure of 0. This creates a special kind of space called a measure space. Because the whole set has a measure of 1, it is also an example of a probability space. This measure can be thought of as modeling something like a fair coin flip, where each outcome has an equal chance.

power set probability space Bernoulli distribution

Important classes of measure spaces

Measure spaces are important in math because of the special properties of their measures. One key type is a probability space, where the measure is a probability measure. Others include finite measure spaces, with a finite measure, and σ-finite measure spaces, where the measure is a σ-finite measure. There are also complete measure spaces.