Measure

• categories: Measure Theory, Definition

A measure is a function that assigns a non-negative value, representing size, length, area, or volume, to subsets of a given space. Formally, a measure $\mu$ on a set $X$ is defined on a $\sigma$-algebra $\mathcal{A}$ of subsets of $X$, satisfying the following properties:

Key Properties:

• Non-negativity: For any $A\in \mathcal{A}$, $\mu (A)\ge 0$.
• Null empty set: $\mu (\varnothing )=0$.
• Countable additivity (σ-additivity): If $\{A_i{\}}_{i=1}^{\infty }$ is a countable collection of disjoint sets in $\mathcal{A}$, then: $$\mu \left(\bigcup _{i=1}^{\infty }{A}_i\right)=\sum _{i=1}^{\infty }\mu (A_i).$$

Common Examples:

• Lebesgue measure: The standard measure on ${\mathbb{R}}^n$, assigning the usual notion of length, area, and volume to intervals and other sets.
• Counting measure: A measure that assigns to each finite set the number of elements it contains.
• Dirac measure: Given a point $x_0\in X$, the Dirac measure ${\delta }_{x_0}$ is defined by ${\delta }_{x_0}(A)=1$ if $x_0\in A$, and ${\delta }_{x_0}(A)=0$ otherwise.

Formal Definition:

A measure $\mu$ on a measurable space $(X,\mathcal{A})$ is a function $\mu :\mathcal{A}\to [0,\infty ]$ such that:

1. $\mu (\varnothing )=0$,
2. For any countable collection of disjoint sets $\{A_i\}\subset \mathcal{A}$,

$$\mu \left(\bigcup _{i=1}^{\infty }{A}_i\right)=\sum _{i=1}^{\infty }\mu (A_i).$$