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 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} is called 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} -finite.
• 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 S} be the collection of all the sets in 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 infinite 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} -measure. 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 \mu} is called semifinite if there exists 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 F \in \mathcal{M}} such 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 F \subseteq E} 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 0 < \mu(F) < + \infty} , for all 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 S} .

Properties

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

1. Finite Additivity: 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 \left\{E_k\right\}_{k = 1}^{n}} be a finite disjoint sequence of sets such that each 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_k \in \mathcal{M}} . 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 \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 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_{n+1} = E_{n+2} = ... = \emptyset} .
2. Monotonicity: 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 E, F \in \mathcal{M}} such 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 E \subseteq F} . 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 \mu\left(E\right) \leq \mu\left(F\right)} .
3. Subadditivity: 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 \left\{E_k\right\}_{k = 1}^{\infty} \subseteq \mathcal{M}} . 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 \mu\left(\cup_{k = 1}^{\infty} E_k\right) \leq \sum_{k = 1}^{\infty} \mu\left(E_k\right)} .
4. Continuity from Below: 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 \left\{E_k\right\}_{k = 1}^{\infty} \subseteq \mathcal{M}} such 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 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 $(X,{\overline {\mathcal {M}}},{\overline {\mu }})$.

References

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