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:

$\mu \left(\emptyset \right)=0$ ,
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.

Types of Measures

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.

Countable 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 $E_{n+1}=E_{n+2}=...=\emptyset$ .
Monotonicity: Let $E,F\in {\mathcal {M}}$ such that $E\subseteq F$ . Then, $\mu \left(E\right)\leq \mu \left(F\right)$ .
Subadditivity: Let $\left\{E_{k}\right\}_{k=1}^{\infty }\subseteq {\mathcal {M}}$ . Then, $\mu \left(\cup _{k=1}^{\infty }E_{k}\right)\leq \sum _{k=1}^{\infty }\mu \left(E_{k}\right)$ .
Continuity from Below: Let $\left\{E_{k}\right\}_{k=1}^{\infty }\subseteq {\mathcal {M}}$ such that $E_{1}\subseteq E_{2}\subseteq ...$ . Then, $\mu \left(\cup _{k=1}^{\infty }E_{k}\right)=\lim _{k\rightarrow \infty }\mu \left(E_{k}\right)$ .
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, $\mu \left(\cap _{k=1}^{\infty }E_{k}\right)=\lim _{k\rightarrow \infty }\mu \left(E_{k}\right)$ .