Measures

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

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

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

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

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

Properties

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

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

Examples

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

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

1. When fixing ${\displaystyle f(x)=1}$, the resulting measure is referred to as the counting measure.
2. Let ${\displaystyle x_{0}\in X}$ be fixed. By defining ${\displaystyle f(x)={\begin{cases}1,x=x_{0}\\0,x\neq x_{0}\end{cases}}}$, the resulting measure is referred to as the point mass measure or the Dirac measure.
• Let ${\displaystyle X}$ be an uncountable set. Let ${\displaystyle {\mathcal {M}}}$ be the ${\displaystyle \sigma }$-algebra of countable or co-countable sets of ${\displaystyle X}$. The function ${\displaystyle \mu :0\rightarrow [0,+\infty ]}$ defined as ${\displaystyle \mu (E)={\begin{cases}0,E{\text{ is countable}}\\1,E{\text{ is co-countable}}\end{cases}}}$ is a measure.
• Let ${\displaystyle X}$ be an infinite set. Let ${\displaystyle {\mathcal {M}}=2^{X}}$. The function ${\displaystyle \mu :0\rightarrow [0,+\infty ]}$ defined as ${\displaystyle \mu (E)={\begin{cases}0,E{\text{ is finite}}\\+\infty ,E{\text{ is infinite}}\end{cases}}}$ is not a measure. To verify that it is not a measure, it is sufficient to take ${\displaystyle X=\mathbb {N} }$, and note that ${\displaystyle \sum _{k=1}^{\infty }\mu \left(\{k\}\right)=0\neq +\infty =\mu \left(\mathbb {N} \right)=\mu \left(\cup _{k=1}^{\infty }\{k\}\right)}$. In other words. the countable additivity property is not satisfied. However, ${\displaystyle \mu }$ does satisfy the finite additivity property.

Complete Measures

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

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

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

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

Borel Measures and Lebesgue Measures

A measure whose domain is the Borel ${\displaystyle \sigma }$-algebra ${\displaystyle {\mathcal {B}}_{\mathbb {R} }}$ is called a Borel measure on ${\displaystyle \mathbb {R} }$. The following theorem provides a method for constructing Borel measures.

Theorem If ${\displaystyle F:\mathbb {R} \rightarrow \mathbb {R} }$ is any increasing, right continuous function, there is a unique Borel measure ${\displaystyle \mu _{F}}$ on ${\displaystyle \mathbb {R} }$ such that ${\displaystyle \mu _{F}((a,b])=F(b)-F(a)}$, for all ${\displaystyle a,b\in \mathbb {R} }$. If ${\displaystyle G:\mathbb {R} \rightarrow \mathbb {R} }$ is another such function, we have ${\displaystyle \mu _{F}=\mu _{G}}$ if and only if ${\displaystyle F-G}$ is constant. Conversely, if ${\displaystyle \mu }$ is a Borel measure on ${\displaystyle \mathbb {R} }$ that is finite on all bounded Borel sets and we define ${\displaystyle F(x)={\begin{cases}\mu ((0,x]),x>0\\0,x=0\\-\mu ((x,0]),x<0\end{cases}}}$, then ${\displaystyle F}$ is increasing and right continuous, and ${\displaystyle \mu =\mu _{F}}$.

A few things should be noted regarding the previous theorem. The ${\displaystyle (a,b]}$ intervals can be replaced by intervals of the form ${\displaystyle [a,b)}$; in this case, the function ${\displaystyle F}$ would have to be left continuous. Additionally, the completion of ${\displaystyle \mu _{F}}$, ${\displaystyle {\overline {\mu _{F}}}}$, is known as the Lebesgue-Stieljes measure associated to ${\displaystyle F}$; this complete measure has a domain that is strictly greater than the ${\displaystyle {\mathcal {B}}_{\mathbb {R} }}$. Finally, taking ${\displaystyle F(x)=x}$ gives rise to the Lebesgue measure.

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.