Measure Theory/Basic Structures And Definitions/Measures

In this section, we study measure spaces and measures.

Measure Spaces
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 measurable 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 measure space

A probability measure is a measure with total measure one (i.e., &mu;(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 $$\mu(E_1) \leq \mu(E_2)$$.

Measures of infinite unions of measurable sets
$$\mu$$ is subadditive: If $$E_1$$, $$E_2$$, $$E_3$$, ... is a countable sequence of sets in $$\Sigma$$, 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
$$\mu$$ 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 n &isin; N, let


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

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:


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