Measure

see also:

• Cardinality vs. Measure
• Divergence (Statistics)

In measure theory, the concept of a “measure” is a fundamental and abstract notion that generalizes the ideas of length, area, volume, and probability. It provides a rigorous mathematical framework for quantifying the size or quantity of a set, particularly within the context of spaces that are more complex than the standard Euclidean Spaces. The development of Measure Theory was a significant advancement in mathematics, enabling precise definitions of integration, probability, and other concepts in analysis.

Definition of a Measure

A measure is a function that assigns a non-negative real number or infinity to subsets of a given space, satisfying certain axioms. Formally, let $X$ be a set and $\mathcal{F}$ a $\sigma$-algebra over $X$ (a collection of subsets of $X$ including the empty set and closed under complementation and Countable unions). A measure $\mu$ is a function $\mu :\mathcal{F}\to [0,\infty ]$ satisfying the following properties:

1. Non-negativity: For every $E\in \mathcal{F}$, $\mu (E)\ge 0$.
2. Null Empty Set: $\mu (\varnothing )=0$.
3. Countable Additivity (or $\sigma$-additivity): For any countable collection $\{E_i{\}}_{i=1}^{\infty }$ of pairwise disjoint sets in $\mathcal{F}$, $\mu \left({\bigcup }_{i=1}^{\infty }{E}_i\right)={\sum }_{i=1}^{\infty }\mu (E_i)$.

The triple $(X,\mathcal{F},\mu )$ is known as a measure space.

Examples of Measures

• Lebesgue Measure: On the real line $\mathbb{R}$, the Lebesgue measure assigns to each interval its length, and this concept is extended to more complex sets in a way that is consistent with our intuitive understanding of length. In higher dimensions, it generalizes to area and volume.

• Counting Measure: Given a set $X$, the counting measure of a subset $E\subseteq X$ is simply the number of elements in $E$ if $E$ is finite, and infinity if $E$ is infinite.

• Probability Measure: In a probability space, the measure (called a probability measure) of a set (an event) is the likelihood of that event occurring, ranging from 0 to 1. This formalizes the concept of probability in mathematical terms.

Importance and Applications

• Integration: The measure provides the foundation for defining the Lebesgue Integral, a generalization of the Riemann integral to a broader class of functions and sets, enabling the integration of functions that are difficult or impossible to integrate using Riemann’s approach.

• Probability Theory: Measure theory underpins modern probability theory, where sets correspond to events and measures to probabilities. This allows for a rigorous mathematical treatment of random variables, expected values, and stochastic processes.

• Functional Analysis: In the context of functional analysis, measures are used to study spaces of functions, such as $L^p$ spaces, which play a critical role in various areas of mathematics and applied fields.

• Ergodic Theory and Dynamical Systems: Measure theory is also essential in ergodic theory, where it is used to analyze the long-term average behavior of dynamical systems.

The concept of measure is central to understanding many areas of mathematics and provides the tools necessary for dealing with continuous phenomena in a precise and logical manner. Its development was a key moment in the formalization of concepts such as integration and probability, and it continues to be a vital area of research in mathematical analysis.