Measure Theory/Basic Structures And Definitions/Measures
In this section, we study measure spaces and measures.
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)"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.
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
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
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 n ∈ N, let
which all have infinite measure, but the intersection is empty.
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.
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.