Measures

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 ${\mathcal {M}}=2^{X}$. The function $\mu :0\rightarrow [0,+\infty ]$ defined as $\mu (E)={\begin{cases}0,E{\text{ is finite}}\\+\infty ,E{\text{ is infinite}}\end{cases}}$ is not a measure. To verify that it is not a measure, it is sufficient to take $X=\mathbb {N}$, and note that $\sum _{k=1}^{\infty }\mu \left(\{k\}\right)=0\neq +\infty =\mu \left(\mathbb {N} \right)=\mu \left(\cup _{k=1}^{\infty }\{k\}\right)$. In other words. the countable additivity property is not satisfied. However, $\mu$ does satisfy the finite additivity property.

Complete Measures

Consider a measure space $(X,{\mathcal {M}},\mu )$. A set $E\in {\mathcal {M}}$ is called a $\mu$-null set (or simply null set) if $\mu (E)=0$. A property $P(x)$ holds $\mu$-almost everywhere (or simply almost everywhere) if $N=\left\{x\in X:P(x){\text{ does not hold}}\right\}$ satisfies $N\in {\mathcal {M}}$ and $\mu (N)=0$.

A measure space $(X,{\mathcal {M}},\mu )$ is called complete if ${\mathcal {M}}$ contains all subsets of its null sets. An incomplete measure space can be constructed by taking $X=\{a.b,c\}$ and ${\mathcal {M}}=\{\emptyset ,\{a\},\{b,c\},X\}$ with $\mu (E)={\begin{cases}0,E\neq \{a\}\\1,E=\{a\}\end{cases}}$. The set $\{b,c\}$ is a null set in this case, but $\{b\}\notin {\mathcal {M}}$.

Given an incomplete measure $(X,{\mathcal {M}},\mu )$, the following theorem guarantees that a complete measure space this measure space can be extended to a complete measure space $(X,{\overline {\mathcal {M}}},{\overline {\mu }})$. The measure ${\overline {\mu }}$ is called the completion of $\mu$, and ${\overline {\mathcal {M}}}$ is called the completion of ${\mathcal {M}}$ with respect to $\mu$.

Theorem Suppose that $(X,{\mathcal {M}},\mu )$ is a measure space. Let ${\mathcal {N}}=\left\{N\in {\mathcal {M}}:\mu (N)=0\right\}$ and ${\overline {\mathcal {M}}}=\left\{E\cup F:E\in {\mathcal {M}}{\text{ and }}F\subseteq N{\text{ for some }}N\in {\mathcal {N}}\right\}$. Then, ${\overline {\mathcal {M}}}$ is a $\sigma$-algebra, and there is a unique extension ${\overline {\mu }}$ of $\mu$ to a complate measure on ${\overline {\mathcal {M}}}$.

References

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