Measure Theory/Basic Structures And Definitions/Measures

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In this section, we study measure spaces and measures.

Measure Spaces[edit]

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 measure 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 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.

Properties[edit]

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

Monotonicity[edit]

\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[edit]

\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) \le \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[edit]

\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 nN, let

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

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

Examples[edit]

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  \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.