Measure space

A measure space is a big idea in measure theory, a part of mathematics. It helps us find the size or amount of different things.

A measure space has three main parts:

• A group of items.
• Some special groups of these items that we can measure (called a σ-algebra).
• A way to measure them (called the measure).

One common example of a measure space is a probability space. This helps us understand chances and likelihoods in games and science.

A related idea is called a measurable space. It has the first two parts—a group of items and groups that can be measured—but it does not have a way to measure them yet. 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 a group 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 look at all possible groups of these items. These groups include having nothing, just 0, just 1, or both 0 and 1 together.

We then decide how to measure these groups. For the groups with just one item (either 0 or 1), we give 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 their special properties. One key type is a probability space, where the measure is a probability measure. Other types include finite measure spaces, which have a finite measure, and σ-finite measure spaces, where the measure is a σ-finite measure. There are also complete measure spaces.