Measures

Measures provide a method for mapping sets to a values in the interval $[0,+\infty ]$. The resulting value can interpreted as the size of the subset. From a geometric perspective, the measure of a set can be viewed as the generalization of length, area, and volume.

Definition

Let $X$ be a set and let ${\mathcal {M}}\subseteq 2^{X}$ be a $\sigma$-algebra. Tbe structure $\left(X,{\mathcal {M}}\right)$ is called a measurable space and each set in ${\mathcal {M}}$ is called a measurable set. A measure on $(X,{\mathcal {M}})$ (also referred to simply as a measure on $X$ if ${\mathcal {M}}$ is understood) is a function $\mu :{\mathcal {M}}\rightarrow [0,+\infty ]$ that satisfies the following criteria:

1. $\mu \left(\emptyset \right)=0$,
2. Let $\left\{E_{k}\right\}_{k=1}^{\infty }$ be a disjoint sequence of sets such that each $E_{k}\in {\mathcal {M}}$. Then, $\mu \left(\cup _{k=1}^{\infty }E_{k}\right)=\sum _{k=1}^{\infty }\mu \left(E_{k}\right)$.

If the previous conditions are satisfied, the structure $\left(X,{\mathcal {M}},\mu \right)$ is called a measure space.

Additional Terminology

Let $\left(X,{\mathcal {M}},\mu \right)$ be a measure space.

• The measure $\mu$ is called finite if $\mu \left(X\right)<+\infty$.
• Let $E\in {\mathcal {M}}$. If there exist $\left\{E_{k}\right\}_{k=1}^{\infty }\subseteq {\mathcal {M}}$ such that $E=\cup _{k=1}^{\infty }E_{k}$ and $\mu \left(E_{k}\right)<+\infty$ (for all $k\in \mathbb {N}$), then $E$ is $\sigma$-finite for $\mu$.
• If $X$ is $\sigma$-finite for $\mu$, then $\mu$ is called $\sigma$-finite.
• Let $S$ be the collection of all the sets in ${\mathcal {M}}$ with infinite $\mu$-measure. The measure $\mu$ is called semifinite if there exists $F\in {\mathcal {M}}$ such that $F\subseteq E$ and $0<\mu (F)<+\infty$, for all $E\in S$.

Properties

Let $\left(X,{\mathcal {M}},\mu \right)$ be a measure space.

1. Finite Additivity: Let $\left\{E_{k}\right\}_{k=1}^{n}$ be a finite disjoint sequence of sets such that each $E_{k}\in {\mathcal {M}}$. Then, $\mu \left(\cup _{k=1}^{n}E_{k}\right)=\sum _{k=1}^{n}\mu \left(E_{k}\right)$. This follows directly from the defintion of measures by taking $E_{n+1}=E_{n+2}=...=\emptyset$.
2. Monotonicity: Let $E,F\in {\mathcal {M}}$ such that $E\subseteq F$. Then, $\mu \left(E\right)\leq \mu \left(F\right)$.
3. Subadditivity: Let $\left\{E_{k}\right\}_{k=1}^{\infty }\subseteq {\mathcal {M}}$. Then, $\mu \left(\cup _{k=1}^{\infty }E_{k}\right)\leq \sum _{k=1}^{\infty }\mu \left(E_{k}\right)$.
4. Continuity from Below: Let $\left\{E_{k}\right\}_{k=1}^{\infty }\subseteq {\mathcal {M}}$ such that $E_{1}\subseteq E_{2}\subseteq ...$. Then, $\mu \left(\cup _{k=1}^{\infty }E_{k}\right)=\lim _{k\rightarrow +\infty }\mu \left(E_{k}\right)$.
5. Continuity from Above: Let $\left\{E_{k}\right\}_{k=1}^{\infty }\subseteq {\mathcal {M}}$ such that $E_{1}\supseteq E_{2}\supseteq ...$ and $\mu \left(E'\right)<+\infty$ for some $E'\in \left\{E_{k}\right\}_{k=1}^{\infty }$. Then, $\mu \left(\cap _{k=1}^{\infty }E_{k}\right)=\lim _{k\rightarrow +\infty }\mu \left(E_{k}\right)$.

Examples

• Let $X$ be a non-empty set and ${\mathcal {M}}=2^{X}$. Let $f$ be any function from $X$ to $[0,+\infty ]$. Given $E\in {\mathcal {M}}$, define $A_{E}=\left\{x\in E:f(x)>0\right\}$. Then, the function $\mu :{\mathcal {M}}\rightarrow [0,+\infty ]$ defined by $\mu (E)={\begin{cases}\sum _{x\in E}f(x),A_{E}{\text{ is countable}}\\+\infty ,A_{E}{\text{ is uncountable}}\end{cases}}$ is a measure. This measure has the following properties:

1. The measure $\mu$ is semifinite if and only if $f(x)<+\infty$ for every $x\in X$.
2. The measure $\mu$ is $\sigma$-finite if and only if $\mu$ is semifinite and $A_{E}$ is countable for every $E\in {\mathcal {M}}$.

There are special cases of this measure that are frequently used:

1. When fixing $f(x)=1$, the resulting measure is referred to as the counting measure.
2. Let $x_{0}\in X$ be fixed. By defining $f(x)={\begin{cases}1,x=x_{0}\\0,x\neq x_{0}\end{cases}}$, the resulting measure is referred to as the point mass measure or the Dirac measure.

• Let $X$ be an uncountable set. Let ${\mathcal {M}}$ be the $\sigma$-algebra of countable or co-cocountable sets of $X$. The function $\mu :0\rightarrow [0,+\infty ]$ defined as $\mu (E)={\begin{cases}0,E{\text{ is countable}}\\1,E{\text{ is co-countable}}\end{cases}}$ is a measure.

• Let $X$ be an infinite set. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{M} = 2^X} . The function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu: 0 \rightarrow [0, +\infty]} defined as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu(E) = \begin{cases}0, E\text{ is finite}\\+\infty, E\text{ is infinite}\end{cases}} is not a measure. To verify that it is not a measure, it is sufficient to take Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X = \mathbb{N}} , and note that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{k = 1}^{\infty} \mu\left(\{k\}\right) = 0 \neq +\infty = \mu\left(\mathbb{N}\right) = \mu\left(\cup_{k = 1}^{\infty} \{k\}\right)} . In other words. the countable additivity property is not satisfied. However, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} does satisfy the finite additivity property.

Complete Measures

Consider a measure space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X, \mathcal{M}, \mu)} . A set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E \in \mathcal{M}} is called a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} -null set (or simply null set) if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu(E) = 0} . A property Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(x)} holds Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} -almost everywhere (or simply almost everywhere) if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N = \left\{x \in X : P(x) \text{ does not hold}\right\}} satisfies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N \in \mathcal{M}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu(N) = 0} .

A measure space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X, \mathcal{M}, \mu)} is called complete if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{M}} contains all subsets of its null sets. An incomplete measure space can be constructed by taking Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X = \{a. b, c\}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{M} = \{\emptyset, \{a\}, \{b, c\}, X\}} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu(E) = \begin{cases}0, E \neq \{a\}\\1, E = \{a\}\end{cases}} . The set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{b, c\}} is a null set in this case, but Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{b\} \notin \mathcal{M}} .

Given an incomplete measure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X, \mathcal{M}, \mu)} , the following theorem guarantees that a complete measure space this measure space can be extended to a complete measure space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X, \overline{\mathcal{M}}, \overline{\mu})} . The measure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{\mu}} is called the completion of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{\mathcal{M}}} is called the completion of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{M}} with respect to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} .

Theorem Suppose that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X, \mathcal{M}, \mu)} is a measure space. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{N} = \left\{N \in \mathcal{M} : \mu(N) = 0\right\}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{\mathcal{M}} = \left\{E \cup F : E \in \mathcal{M} \text{ and } F \subseteq N \text{ for some } N \in \mathcal{N}\right\}} . Then, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{\mathcal{M}}} is a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} -algebra, and there is a unique extension Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{\mu}} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} to a complate measure on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{\mathcal{M}}} .

Borel and Lebesgue Measures

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References

1. Folland, Gerald B. (1999). Real Analysis: Modern Techniques and Their Applications, John Wiley and Sons, ISBN 0471317160, Second edition.
2. Craig, Katy. MATH 201A Lectures 4-5, 7-8. UC Santa Barbara, Fall 2020.