Measure Theory/Basic Structures And Definitions/Measures

In this section, we study measure spaces and measures.

Measure Spaces

Let ${\displaystyle X}$  be a set and ${\displaystyle {\mathcal {M}}}$  be a collection of subsets of ${\displaystyle X}$  such that ${\displaystyle {\mathcal {M}}}$  is a σ-ring.

We call the pair ${\displaystyle \left\langle X,{\mathcal {M}}\right\rangle }$  a measurable space. Members of ${\displaystyle {\mathcal {M}}}$  are called measurable sets.

A positive real valued function ${\displaystyle \mu }$  defined on ${\displaystyle {\mathcal {M}}}$  is said to be a measure if and only if,

(i)${\displaystyle \mu (\varnothing )=0}$  and

(i)"Countable additivity": ${\displaystyle \mu \left(\bigcup _{i=1}^{\infty }E_{i}\right)=\sum _{i=1}^{\infty }\mu (E_{i})}$ , where ${\displaystyle E_{i}\in {\mathcal {M}}}$  are pairwise disjoint sets.

we call the triplet ${\displaystyle \left\langle X,{\mathcal {M}},\mu \right\rangle }$  a measure space

A probability measure is a measure with total measure one (i.e., μ(X)=1); a probability space is a measure space with a probability measure.

Properties

Several further properties can be derived from the definition of a countably additive measure.

Monotonicity

${\displaystyle \mu }$  is monotonic: If ${\displaystyle E_{1}}$  and ${\displaystyle E_{2}}$  are measurable sets with ${\displaystyle E_{1}\subseteq E_{2}}$  then ${\displaystyle \mu (E_{1})\leq \mu (E_{2})}$ .

Measures of infinite unions of measurable sets

${\displaystyle \mu }$  is subadditive: If ${\displaystyle E_{1}}$ , ${\displaystyle E_{2}}$ , ${\displaystyle E_{3}}$ , ... is a countable sequence of sets in ${\displaystyle \Sigma }$ , not necessarily disjoint, then

${\displaystyle \mu \left(\bigcup _{i=1}^{\infty }E_{i}\right)\leq \sum _{i=1}^{\infty }\mu (E_{i})}$ .

${\displaystyle \mu }$  is continuous from below: If ${\displaystyle E_{1}}$ , ${\displaystyle E_{2}}$ , ${\displaystyle E_{3}}$ , ... are measurable sets and ${\displaystyle E_{n}}$  is a subset of ${\displaystyle E_{n+1}}$  for all n, then the union of the sets ${\displaystyle E_{n}}$  is measurable, and

${\displaystyle \mu \left(\bigcup _{i=1}^{\infty }E_{i}\right)=\lim _{i\to \infty }\mu (E_{i})}$ .

Measures of infinite intersections of measurable sets

${\displaystyle \mu }$  is continuous from above: If ${\displaystyle E_{1}}$ , ${\displaystyle E_{2}}$ , ${\displaystyle E_{3}}$ , ... are measurable sets and ${\displaystyle E_{n+1}}$  is a subset of ${\displaystyle E_{n}}$  for all n, then the intersection of the sets ${\displaystyle E_{n}}$  is measurable; furthermore, if at least one of the ${\displaystyle E_{n}}$  has finite measure, then

${\displaystyle \mu \left(\bigcap _{i=1}^{\infty }E_{i}\right)=\lim _{i\to \infty }\mu (E_{i})}$ .

This property is false without the assumption that at least one of the ${\displaystyle E_{n}}$  has finite measure. For instance, for each nN, let

${\displaystyle E_{n}=[n,\infty )\subseteq \mathbb {R} }$

which all have infinite measure, but the intersection is empty.

Counting Measure

Start with a set Ω and consider the sigma algebra X on Ω consisting of all subsets of Ω. Define a measure μ on this sigma algebra by setting μ(A) = |A| if A is a finite subset of Ω and μ(A) = ∞ if A is an infinite subset of Ω, where |A| denotes the cardinality of set A. Then (Ω, X, μ) is a measure space. μ is called the counting measure.

Lebesgue Measure

For any subset B of Rn, we can define an outer measure ${\displaystyle \lambda ^{*}}$  by:

${\displaystyle \lambda ^{*}(B)=\inf\{\operatorname {vol} (M):M\supseteq B\}}$ , and ${\displaystyle M\ }$  is a countable union of products of intervals .

Here, vol(M) is sum of the product of the lengths of the involved intervals. We then define the set A to be Lebesgue measurable if

${\displaystyle \lambda ^{*}(B)=\lambda ^{*}(A\cap B)+\lambda ^{*}(B-A)}$

for all sets B. These Lebesgue measurable sets form a σ-algebra, and the Lebesgue measure is defined by λ(A) = λ*(A) for any Lebesgue measurable set A.