Measure Theory

A measurable space is a space on which a measure can be defined. It consists of a pair $(X,\mathcal{F})$ where $X$ is a set and $\mathcal{F}$ is a sigma algebra on $X$ . Each set $A \in \mathcal{F}$ is called a measurable set. A measure is a non-negative and countably additive function $\mu$ defined on $\mathcal{F}$ such that $\mu(\emptyset) = 0$ . Countable additivity is the statement that the measure of a countable union of mutually disjoint measurable sets is the sum of the measures of the sets comprising the union.

The purpose of measure is to capture essential properties of notions such as length and area to more general situations. Given a set of disjoint intervals and a ruler (the measure function) we can calculate the length of an interval (its measure) . Given a countable set of intervals that have no overlap, the measure of the set should be the sum of the lengths of each interval in the set. This suggests that measure should be translation invariant, that is the length of an interval does not change when the interval is translated. It can be established that a countably additive translation invariant that generalizes length cannot be consistently defined on all sets of the real line, hence the technical assumption that the measurable sets be restricted to sigma algebras. In the case of the real line, the sigma algebra corresponds to the sigma algebra generated from the open intervals, also called the Borel algebra.

A probability measure is a measure for which $\mu(X) = 1$ .

Some Properties fo Measures

lemma: measures are finitely additive.

proof: Let $A_1,A_2$ be disjoint measurable sets. extend the list to a sequence of measurable sets such that $A_{i + 2} = \emptyset$ when $i = 1,2,\ldots$ . Then by countable additivity,

$\mu(A_1 \cup A_2) = \mu(A_1) + \mu(A_2) + \sum_{i = 1,\ldots} \mu \emptyset = \mu(A_1) + \mu(A_2)$ .

By induction, the general case can be established.

lemma: measures are non-decreasing.

Let $A,B$ be measurable sets such that $A \subseteq B$ . Since $B = A \cup (B-A)$ is the disjoint union of measurable sets, $\mu(B) = \mu(A) + \mu(B-A) \ge \mu(A)$ .

lemma: measures are subadditive.

let $A_1,A_2,\ldots$ be a sequence of measurable sets. Then the sequence $B$ defined by $B_i = A_i -\cup_{j=1..i-1} A_j$ for $i = 1,2,\ldots$ is a disjoint sequence of measurable sets such that $\cup_i A_i = \cup_i B_i$ . Hence

$\mu(\cup_i A_i) = \sum_i \mu(B_i) \le \sum_i \mu(A_i)$

lemma: the measure of the limit of an increasing sequence of measurable set is the limit of the sequence of measures of the measurable sets.

Let $A$ be a non-decreasing sequence of measurable sets. Then $\cup_i A_i = A_1 \cup (A_2 - A_1) \cup \ldots$ can be expressed as a disjoint union of annular measurable sets. Hence

$\mu(\cup_i A_i) = \mu A_1 + \sum_i \mu(A_{i+1} - A_i) = \mu A_1 + \lim_{n \rightarrow \infty} \sum_{i=1..n} \mu(A_{i+1} - A_i)$ .

Either the measure of all sets are finite, or there is a set $A_i$ whose measure is infinite. In the latter case, by monotonicity, $\mu(\cup_i A_i) \ge \mu(A_{i+j}) \ge \mu(A_i) = \infty$ and so $\mu(\cup_i A_i) = \lim_i \mu(A_i) = \infty$ . Otherwise, $\mu(A_{i+1} - A_i) = \mu(A_{i+1}) - \mu(A_i)$ and since the sum of differences is the difference of the endpoints,

$\mu(\cup_i A_i) = \mu A_1 + \lim_{n \rightarrow \infty} \mu A_{n+1} - \mu A_1 = \lim_{n \rightarrow \infty} \mu A_n$

lemma: the measure of the limit of a non-increasing sequence of measurable sets of finite measure is the limit of the sequence of measures of the sets.

Let $A$ be a non-increasing sequence of measurable sets. Then the sequence $B$ defined by $B_i = A_1 - A_i$ for $i = 1,2,\ldots$ is a non-decreasing sequence of measurable sets. Hence

$\mu(\cup B_i) = \lim_{n \rightarrow \infty} \mu(B_n)$ .

Since $\mu(\cup_i B_i) = \mu(A_1) - \mu(\cap_i A_i)$ and $\mu(B_n) = \mu(A_1) - \mu(A_n)$ the conclusion

$\mu(\cap_i A_i) = \lim_{n \rightarrow \infty} \mu(A_n)$

example: Let $X = \{ x_i : i \in \mathbb{N}\}$ be a countable set, and let $p$ be a non-negative function on $X$ such that $\sum_i p(x_i) = 1$ . Then for any subset $A$ of $X$ defining the set function $PA = \sum_{i : x_i \in A} p(x_i)$ . Then $P$ is a probability measure on $2^X$ .

lemma: the intersection of the collection of sigma algebras of $X$ that contain a collection of subsets of $X$ is a sigma algebra of $X$ .

Note that the intersection of a collection of sets that contains a collection $C$ also contains the collection $C$ . Let $A$ be a set in the intersection of the collection of sigma algebras containing $C$ . Then it is a set of every sigma algebra in the collection, and so $A^C$ is also a set of every sigma algebra in the collection, and so is in the intersection of the collection. Let $\left<A_i\right>$ be a sequence of sets in the collection of sigma algebras. Then it is also a sequence in every sigma algebra of the collection, and by countable additivity $\cup_i A_i$ is an element in every sigma algebra of the collection, and so belongs to the intersection of the collection. Finally, since $X$ is an element of every sigma algebra of $X$ , it follows it belongs to the intersection of the collection. Hence the intersection is a sigma algebra of $X$ containing $C$ .

By definition, the intersection is contained in every sigma algebra in the collection of sigma algebras containing $C$ and so is the smallest such sigma algebra.

This notion enable one to generate sigma algebras that ensure all sets in given collection are measurable sets. Of importance are the sigma algebras generated from open intervals of the real line or its multidimensional analogues. These are called Borel sets (of dimension $d$ ) where $d = 1,2\ldots$ . Borel sets are general enough to capture area of enclosed continuous curves in planar space, for example, since every region enclosed by a continuous curve is the countable union of disjoint open rectangles.

It remains to provide a means of constructing measures. Using the real line as the starting basis for analysis, define

definition: a Stieltjes function is a non-decreasing and right-continuous real-valued function on the real line.

theorem: associated with each Stieltjes function $F$ is a unique measure on Borel sets such that $\mu( (a,b]) = F(b) - F(a)$ .

The proof is very intricate and relies on non-intutive theory. It will not be discussed in depth at this juncture. However some essential details will be described.

The simplest such function is $F(x) = x$ , where the measure of an interval corresponds with its length.

definition: A semi-algebra on a set $X$ is a collection of subsets of $X$ that is closed under pairwise set intersection and such that the complement of any set in the collection is a finite union of sets in the collection.

example: Consider the collection of $n$ -dimensional rectangles

$(a_1,b_1] \times (a_2,b_2] \times \ldots (a_n,b_n]$

This collection is a semi-algebra on $\mathbb{R}^n$ .

definition: an algebra of a set $X$ is a collection of subsets of $X$ that is closed under pairwise set intersection and set complement.

lemma: if $S$ is a semi-algebra on $X$ , then the collection of all finite unions of sets of $S$ is an algebra.

proof: skip

definition: a measure on an algebra $A$ of the set $X$ is a non-negative set function $\mu$ on $A$ such that $\mu \emptyset = 0$ and given any countable disjoint collection of sets in the algebra such that the union is also a set in the algebra, the measure of the union is the sum of the measures of the sets comprising the union:

$\mu \cup_i A_i = \sum_i \mu A_i$ .

definition: a measure $\mu$ on an algebra of $X$ is $\sigma$ -finite if there is a non-decreasing sequence of sets of finite measure whose union covers $X$ .

The following theorem allows one to extend the domain of definition of appropriately defined functions from a semi-algebra to a sigma algebra.

theorem: Let $S$ be a semialgebra on $X$ and $\mu$ be a non-negative function defined on $S$ and such that $\mu \emptyset= 0$ . If the following two conditions below are satsified, then $\mu$ has a unique extension $\mu_2$ to the algebra generated from $S$ .

the measure of the union of a finite disjoint sequence of sets in the semialgebra is the sum of the measures of the sets in the sequence
the measure of the union of a countable disjoint sequence of sets in the semialgebra is not larger than the sum of the measures of the sets in the sequence.

If the extension $\mu_2$ is $\sigma$ -finite then there is an extension $\mu_3$ of the measure to the $\sigma$ -algebra containing $S$ .

Measure Theory

Some Properties fo Measures

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