Measure (mathematics)

In mathematics, a measure is a way to give a size, length, area, or volume to different things. It helps us understand how big, small, heavy, or likely something is. Measures are used in many areas, like figuring out the chance of events happening in probability theory or adding up areas under curves in integration theory.

The idea of measuring things goes back a long time, even to Ancient Greece, when smart people like Archimedes tried to find the area of a circle. But it wasn’t until the late 1800s and early 1900s that measure became a special part of math. Great mathematicians like Émile Borel, Henri Lebesgue, and Constantin Carathéodory helped build the rules we use today.

Measures can even be used in surprising places, like quantum physics, where they help describe things we can’t see with our eyes. Whether you’re measuring the length of a room, the volume of water, or the probability of rain, measures help us make sense of the world.

Definition

A measure is a way to assign a number to sets of things, like how long something is or how much space it takes up. In math, we use measures to talk about sizes, amounts, and even chances.

For example, the measure of a line might be its length, and the measure of a shape might be its area. Measures help us compare and work with these ideas in a clear and consistent way.

Instances

Main category: Measures (measure theory)

Some important measures are listed here:

• The counting measure tells us how many items are in a group.
• The Lebesgue measure helps us understand lengths, areas, and volumes in a very general way.
• The arc length of an interval on the unit circle can be used as a measure of angles.
• The Haar measure is important for studying groups in mathematics.
• Every probability space gives us a measure that shows how likely different events are.
• The Dirac measure focuses on a single point, assigning the value 1 to any set that includes that point and 0 to sets that do not.

Other named measures used in various theories include: Borel measure, Jordan measure, ergodic measure, Gaussian measure, Baire measure, Radon measure, Young measure, and Loeb measure.

In physics, measures can describe how mass or other properties are spread out in space.

Basic properties

In mathematics, a measure helps us understand size, like length or area, but in a more general way. It can apply to many different things, not just shapes.

One key idea is that if you have two sets where one is completely inside the other, the measure of the smaller set will always be less than or equal to the measure of the larger set. Another important idea is how measures behave when you combine many sets together or look at their overlaps. These properties help mathematicians use measures in many areas, like probability and calculus.

Main article: Measure (mathematics))

Other properties

Completeness

Main article: Complete measure

A measurable set (X) is called a null set if (\mu(X) = 0). A subset of a null set is called a negligible set. A measure is called complete if every negligible set is measurable.

"Dropping the edge"

If (f: X \to [0, +\infty]) is measurable, then (\mu{x \in X : f(x) \geq t} = \mu{x \in X : f(x) > t}) for almost all (t \in [-\infty, \infty]). This property is used in connection with the Lebesgue integral.

Additivity

Measures are required to be countably additive. This condition can be strengthened in certain ways for specific types of measures.

Finite and σ-finite measures

A measure space ((X, \Sigma, \mu)) is called finite if (\mu(X)) is a finite real number. A measure (\mu) is called σ-finite if (X) can be decomposed into a countable union of measurable sets of finite measure.

For example, the real numbers with the standard Lebesgue measure are σ-finite but not finite. The natural numbers are also σ-finite with respect to counting measure.

Strictly localizable measures

Main article: Decomposable measure

Semifinite measures

Let (X) be a set, let (\mathcal{A}) be a sigma-algebra on (X), and let (\mu) be a measure on (\mathcal{A}). We say (\mu) is semifinite if for all (A \in \mu^{\text{pre}}{+\infty}), (\mathcal{P}(A) \cap \mu^{\text{pre}}(\mathbb{R}_{>0}) \neq \emptyset).

Semifinite measures generalize sigma-finite measures, allowing some big theorems of measure theory to be extended with little modification.

Localizable measures

Localizable measures are a special case of semifinite measures and a generalization of sigma-finite measures.

s-finite measures

Main article: s-finite measure

A measure is said to be s-finite if it is a countable sum of finite measures. S-finite measures are more general than sigma-finite ones and have applications in the theory of stochastic processes.

Non-measurable sets

Main article: Non-measurable set

In mathematics, some special sets cannot be measured using standard rules. This happens when we assume something called the "axiom of choice" to be true. Examples of such tricky sets include the Vitali set, and ideas from the Hausdorff paradox and the Banach–Tarski paradox. These sets show that not every group of points in space can have a clear size or measure.

Generalizations

In mathematics, a measure is a way to assign values like length or area to objects. Sometimes, these values can be more complex than just positive numbers. For example, a signed measure can have positive or negative values, and a complex measure uses complex numbers.

There are also special types of measures, like projection-valued measures, which are used in advanced areas of mathematics. Some measures only need to add up for a small number of objects at a time, rather than many, and these are called finitely additive measures. These ideas help solve tricky problems in geometry and other areas of math.