Measure Theory/Basic Structures And Definitions/Measures

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In this section, we study measure spaces and measures.

Measure Spaces[edit]

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

We call the pair a measure 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 measurable 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.


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


is monotonic: If and are measurable sets with then .

Measures of infinite unions of measurable sets[edit]

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 sets[edit]

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.


Counting Measure[edit]

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[edit]

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.