Measure Theory/Basic Structures And Definitions/Measures

In this section, we study measure spaces and measures.

Measure Spaces

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

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

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

(i)$\mu (\varnothing )=0$ and

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

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

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

Measures of infinite unions of measurable sets

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

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

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

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

Measures of infinite intersections of measurable sets

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

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

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

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

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 $\lambda ^{*}$ by:

$\lambda ^{*}(B)=\inf\{\operatorname {vol} (M):M\supseteq B\}$, and $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

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

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.