Measure Theory/Basic Structures And Definitions/Measures

In this section, we study measure spaces and measures.

Measure SpacesEdit

Let   be a set and   be a collection of subsets of   such that   is a σ-ring.

We call the pair   a measurable space. Members of   are called measurable sets.

A positive real valued function   defined on   is said to be a measure if and only if,

(i)  and

(i)"Countable additivity":  , where   are pairwise disjoint sets.

we call the triplet   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.

PropertiesEdit

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

MonotonicityEdit

  is monotonic: If   and   are measurable sets with   then  .

Measures of infinite unions of measurable setsEdit

  is subadditive: If  ,  ,  , ... is a countable sequence of sets in  , not necessarily disjoint, then

 .

  is continuous from below: If  ,  ,  , ... are measurable sets and   is a subset of   for all n, then the union of the sets   is measurable, and

 .

Measures of infinite intersections of measurable setsEdit

  is continuous from above: If  ,  ,  , ... are measurable sets and   is a subset of   for all n, then the intersection of the sets   is measurable; furthermore, if at least one of the   has finite measure, then

 .

This property is false without the assumption that at least one of the   has finite measure. For instance, for each nN, let

 

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

ExamplesEdit

Counting MeasureEdit

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 MeasureEdit

For any subset B of Rn, we can define an outer measure   by:

 , and   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

 

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.