Measure Theory/Basic Structures And Definitions/Measures

Formally, a measure μ is a function defined on a σ-algebra Σ over a set X and taking values in the extended interval [0,∞] such that the following properties are satisfied:

• The empty set has measure zero:

$\mu (\varnothing )=0$.

• Countable additivity or σ-additivity'': If $E_1,E_2,E_3,$ ... is a countable sequence of pairwise disjoint sets in $\mathrm{\Sigma }$, the measure of the union of all the $E_i$'s is equal to the sum of the measures of each $E_i$:

$\mu \left(\bigcup _{i=1}^{\mathrm{\infty }}{E}_i\right)=\sum _{i=1}^{\mathrm{\infty }}\mu (E_i).$

The triple (X,Σ,μ) is then called a measure space, and the members of Σ are called measurable sets.

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 $\mu (E_1)\le \mu (E_2)$.

Measures of infinite unions of measurable sets

$\mu$ is subadditive: If $E_1$, $E_2$, $E_3$, ... is a countable sequence of sets in $\mathrm{\Sigma }$, not necessarily disjoint, then

$\mu \left(\bigcup _{i=1}^{\mathrm{\infty }}{E}_i\right)\le \sum _{i=1}^{\mathrm{\infty }}\mu (E_i)$.

$\mu$ 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

$\mu \left(\bigcup _{i=1}^{\mathrm{\infty }}{E}_i\right)=\underset{i\to \mathrm{\infty }}{\lim }\mu (E_i)$.

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

$\mu \left(\bigcap _{i=1}^{\mathrm{\infty }}{E}_i\right)=\underset{i\to \mathrm{\infty }}{\lim }\mu (E_i)$.

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

$E_n=[n,\mathrm{\infty })\subseteq ℝ$

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:

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