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Measures provide a method for mapping set to a value in the interval <math>[0, +\infty]</math>. The resulting value can interpreted as the size of the subset. From a geometric perspective, the measure of a set can be viewed as the generalization of length, area, and volume.


==Definition==
==Definition==
Let <math>X</math> be a set and let <math>\mathcal{M} \subseteq 2^X</math> be a <math>\sigma</math>-algebra. Tbe structure <math>\left(X, \mathcal{M}\right)</math> is called a '''measurable space''' and each set in <math>\mathcal{M}</math> is called a '''measurable set'''. A '''measure on <math>(X, \mathcal{M})</math>''' (also referred to simply as a '''measure on <math>X</math>''' if <math>\mathcal{M}</math> is understood) is a function <math>\mu: \mathcal{M} \rightarrow [0, +\infty]</math> that satisfies the following criteria:
# <math>\mu\left(\emptyset\right) = 0</math>,
# Let <math>\left\{E_k\right\}_{k = 1}^{\infty}</math> be a disjoint sequence of sets such that each <math>E_k \in \mathcal{M}</math>. Then, <math>\mu\left(\cup_{k = 1}^{\infty} E_k\right) = \sum_{k = 1}^{\infty} \mu\left(E_k\right)</math>.
If the previous conditions are satisfied, the structure <math>\left(X, \mathcal{M}, \mu\right)</math> is called a '''measure space'''.
==Additional Terminology==
Let <math>\left(X, \mathcal{M}, \mu\right)</math> be a measure space.
* The measure <math>\mu</math> is called '''finite''' if <math>\mu\left(X\right) < +\infty</math>.
* Let <math>E \in \mathcal{M}</math>. If there exist <math>\left\{E_k\right\}_{k = 1}^{\infty} \subseteq \mathcal{M}</math>  such that <math>E = \cup_{k = 1}^{\infty} E_k</math> and <math>\mu\left(E_k\right) < + \infty</math> (for all <math>k \in \mathbb{N}</math>), then <math>E</math> is '''<math>\sigma</math>-finite for <math>\mu</math>'''.
* If <math>X</math> is <math>\sigma</math>-finite for <math>\mu</math>, then <math>\mu</math> is called '''<math>\sigma</math>-finite'''.
* Let <math>S</math> be the collection of all the sets in <math>\mathcal{M}</math> with infinite <math>\mu</math>-measure. The measure <math>\mu</math> is called '''semifinite''' if there exists <math>F \in \mathcal{M}</math> such that <math>F \subseteq E</math> and <math>0 < \mu(F) < + \infty</math>, for all <math>E \in S</math>.
==Properties==
Let <math>\left(X, \mathcal{M}, \mu\right)</math> be a measure space.
# '''Finite Additivity:''' Let <math>\left\{E_k\right\}_{k = 1}^{n}</math> be a finite disjoint sequence of sets such that each <math>E_k \in \mathcal{M}</math>. Then, <math>\mu\left(\cup_{k = 1}^{n} E_k\right) = \sum_{k = 1}^{n} \mu\left(E_k\right)</math>. This follows directly from the defintion of measures by taking <math>E_{n+1} = E_{n+2} = ... = \emptyset</math>.
# '''Monotonicity:''' Let <math>E, F \in \mathcal{M}</math> such that <math>E \subseteq F</math>. Then, <math>\mu\left(E\right) \leq \mu\left(F\right)</math>.
# '''Subadditivity:''' Let <math>\left\{E_k\right\}_{k = 1}^{\infty} \subseteq \mathcal{M}</math>. Then, <math>\mu\left(\cup_{k = 1}^{\infty} E_k\right) \leq \sum_{k = 1}^{\infty} \mu\left(E_k\right)</math>.
# '''Continuity from Below:''' Let <math>\left\{E_k\right\}_{k = 1}^{\infty} \subseteq \mathcal{M}</math> such that <math>E_1 \subseteq E_2 \subseteq ...</math>. Then, <math>\mu\left(\cup_{k = 1}^{\infty} E_k\right) = \lim_{k \rightarrow +\infty} \mu\left(E_k\right)</math>.
# '''Continuity from Above:''' Let <math>\left\{E_k\right\}_{k = 1}^{\infty} \subseteq \mathcal{M}</math> such that <math>E_1 \supseteq E_2 \supseteq ...</math> and <math>\mu\left(E'\right) < +\infty</math> for some <math>E' \in \left\{E_k\right\}_{k = 1}^{\infty}</math>. Then, <math>\mu\left(\cap_{k = 1}^{\infty} E_k\right) = \lim_{k \rightarrow +\infty} \mu\left(E_k\right)</math>.
==Examples==
* Let <math>X</math> be a non-empty set and <math>\mathcal{M} = 2^X</math>. Let <math>f</math> be any function from <math>X</math> to <math>[0, +\infty]</math>. Given <math>E \in \mathcal{M}</math>, define <math>A_E = \left\{x \in E : f(x) > 0\right\}</math>. Then, the function <math>\mu: \mathcal{M} \rightarrow [0, +\infty]</math> defined by <math>\mu(E) = \begin{cases}\sum_{x \in E}f(x), A_E\text{ is countable}\\+\infty, A_E\text{ is uncountable}\end{cases}</math> is a measure. This measure has the following properties:
# The measure <math>\mu</math> is semifinite if and only if <math>f(x) < +\infty</math> for every <math>x \in X</math>.
# The measure <math>\mu</math> is <math>\sigma</math>-finite if and only if <math>\mu</math> is semifinite and <math>A_E</math> is countable for every <math>E \in \mathcal{M}</math>.
There are special cases of this measure that are frequently used:
# When fixing <math>f(x) = 1</math>, the resulting measure is referred to as the '''counting measure'''.
# Let <math>x_0 \in X</math> be fixed. By defining <math>f(x) = \begin{cases}1, x = x_0\\0, x \neq x_0\end{cases}</math>, the resulting measure is referred to as the '''point mass measure''' or the '''Dirac measure'''.
* Let <math>X</math> be an uncountable set. Let <math>\mathcal{M}</math> be the <math>\sigma</math>-algebra of countable or co-countable sets of <math>X</math>. The function <math>\mu: 0 \rightarrow [0, +\infty]</math> defined as <math>\mu(E) = \begin{cases}0, E\text{ is countable}\\1, E\text{ is co-countable}\end{cases}</math> is a measure.
* Let <math>X</math> be an infinite set. Let <math>\mathcal{M} = 2^X</math>. The function <math>\mu: 0 \rightarrow [0, +\infty]</math> defined as <math>\mu(E) = \begin{cases}0, E\text{ is finite}\\+\infty, E\text{ is infinite}\end{cases}</math> is not a measure. To verify that it is not a measure, it is sufficient to take <math>X = \mathbb{N}</math>, and note that <math>\sum_{k = 1}^{\infty} \mu\left(\{k\}\right) = 0 \neq +\infty = \mu\left(\mathbb{N}\right) = \mu\left(\cup_{k = 1}^{\infty} \{k\}\right)</math>. In other words. the countable additivity property is not satisfied. However, <math>\mu</math> does satisfy the finite additivity property.
==Complete Measures==
Consider a measure space <math>(X, \mathcal{M}, \mu)</math>. A set <math>E \in \mathcal{M}</math> is called a <math>\mu</math>-'''null set''' (or simply '''null set''') if <math>\mu(E) = 0</math>. A property <math>P(x)</math> holds <math>\mu</math>-'''almost everywhere''' (or simply '''almost everywhere''') if <math>N = \left\{x \in X : P(x) \text{ does not hold}\right\}</math> satisfies <math>N \in \mathcal{M}</math> and <math>\mu(N) = 0</math>.
A measure space <math>(X, \mathcal{M}, \mu)</math> is called '''complete''' if <math>\mathcal{M}</math> contains all subsets of its null sets. An incomplete measure space can be constructed by taking <math>X = \{a. b, c\}</math> and <math>\mathcal{M} = \{\emptyset, \{a\}, \{b, c\}, X\}</math> with <math>\mu(E) = \begin{cases}0, E \neq \{a\}\\1, E = \{a\}\end{cases}</math>. The set <math>\{b, c\}</math> is a null set in this case, but <math>\{b\} \notin \mathcal{M}</math>.
Given an incomplete measure <math>(X, \mathcal{M}, \mu)</math>, the following theorem guarantees that a complete measure space this measure space can be extended to a complete measure space <math>(X, \overline{\mathcal{M}}, \overline{\mu})</math>. The measure <math>\overline{\mu}</math> is called the '''completion of <math>\mu</math>''', and <math>\overline{\mathcal{M}}</math> is called the '''completion of <math>\mathcal{M}</math> with respect to <math>\mu</math>'''.
'''Theorem''' <em>Suppose that <math>(X, \mathcal{M}, \mu)</math> is a measure space. Let <math>\mathcal{N} = \left\{N \in \mathcal{M} : \mu(N) = 0\right\}</math> and <math>\overline{\mathcal{M}} = \left\{E \cup F : E \in \mathcal{M} \text{ and } F \subseteq N  \text{ for some } N \in \mathcal{N}\right\}</math>. Then, <math>\overline{\mathcal{M}}</math> is a <math>\sigma</math>-algebra, and there is a unique extension <math>\overline{\mu}</math> of <math>\mu</math> to a complate measure on <math>\overline{\mathcal{M}}</math>.</em>


Let <math>X</math> be a set equipped with a <math>\sigma</math>-algebra <math>\mathcal{M}</math>. A '''measure on <math>(X, \mathcal{M})</math>''' (also referred to simply as '''measure on <math>X</math>''' if <math>M</math> is understood) is a function <math>\mu: \mathcal{M} \rightarrow [0, \infty]</math> that satisfies the following criteria:
==Borel Measures and Lebesgue Measures==
# <math>\mu\left(\emptyset\right) = 0</math>,
A measure whose domain is the Borel <math>\sigma</math>-algebra <math>\mathcal{B}_\mathbb{R}</math> is called a '''Borel measure on <math>\mathbb{R}</math>'''. The following theorem provides a method for constructing Borel measures.
# Let <math>\left\{E_k\right\}_{k = 1}^{\infty}</math> be a disjoint sequence of sets contained in <math>\mathcal{M}</math>. Then, <math>\mu\left(\cup_{k = 1}^{\infty} E_k\right) = 0</math>.
 
'''Theorem''' <em>If <math>F: \mathbb{R} \rightarrow \mathbb{R}</math> is any increasing, right continuous function, there is a unique Borel measure <math>\mu_F</math> on <math>\mathbb{R}</math> such that <math>\mu_F((a, b]) = F(b) - F(a)</math>, for all <math>a, b \in \mathbb{R}</math>. If <math>G: \mathbb{R} \rightarrow \mathbb{R}</math> is another such function, we have <math>\mu_F = \mu_G</math> if and only if <math>F - G</math> is constant. Conversely, if <math>\mu</math> is a Borel measure on <math>\mathbb{R}</math> that is finite on all bounded Borel sets and we define <math>F(x) = \begin{cases}\mu((0, x]), x > 0\\0, x = 0\\-\mu((x, 0]), x < 0\end{cases}</math>, then <math>F</math> is increasing and right continuous, and <math>\mu = \mu_F</math>.</em>
 
A few things should be noted regarding the previous theorem. The <math>(a, b]</math> intervals can be replaced by intervals of the form <math>[a, b)</math>; in this case, the function <math>F</math> would have to be left continuous. Additionally, the completion of <math>\mu_F</math>, <math>\overline{\mu_F}</math>, is known as the '''[[Lebesgue-Stieljes_Measures|Lebesgue-Stieljes measure]] associated to <math>F</math>'''; this complete measure has a domain that is strictly greater than the <math>\mathcal{B}_\mathbb{R}</math>. Finally, taking <math>F(x) = x</math> gives rise to the Lebesgue measure.
 
==References==
# Folland, Gerald B. (1999). Real Analysis: Modern Techniques and Their Applications, John Wiley and Sons, ISBN 0471317160, Second edition.
# Craig, Katy. MATH 201A Lectures 4-5, 7-8. UC Santa Barbara, Fall 2020.

Latest revision as of 04:14, 19 December 2020

Measures provide a method for mapping set to a value in the interval . The resulting value can interpreted as the size of the subset. From a geometric perspective, the measure of a set can be viewed as the generalization of length, area, and volume.

Definition

Let be a set and let be a -algebra. Tbe structure is called a measurable space and each set in is called a measurable set. A measure on (also referred to simply as a measure on if is understood) is a function that satisfies the following criteria:

  1. ,
  2. Let be a disjoint sequence of sets such that each . Then, .

If the previous conditions are satisfied, the structure is called a measure space.

Additional Terminology

Let be a measure space.

  • The measure is called finite if .
  • Let . If there exist such that and (for all ), then is -finite for .
  • If is -finite for , then is called -finite.
  • Let be the collection of all the sets in with infinite -measure. The measure is called semifinite if there exists such that and , for all .

Properties

Let be a measure space.

  1. Finite Additivity: Let be a finite disjoint sequence of sets such that each . Then, . This follows directly from the defintion of measures by taking .
  2. Monotonicity: Let such that . Then, .
  3. Subadditivity: Let . Then, .
  4. Continuity from Below: Let such that . Then, .
  5. Continuity from Above: Let such that and for some . Then, .

Examples

  • Let be a non-empty set and . Let be any function from to . Given , define . Then, the function defined by is a measure. This measure has the following properties:
  1. The measure is semifinite if and only if for every .
  2. The measure is -finite if and only if is semifinite and is countable for every .

There are special cases of this measure that are frequently used:

  1. When fixing , the resulting measure is referred to as the counting measure.
  2. Let be fixed. By defining , the resulting measure is referred to as the point mass measure or the Dirac measure.
  • Let be an uncountable set. Let be the -algebra of countable or co-countable sets of . The function defined as is a measure.
  • Let be an infinite set. Let . The function defined as is not a measure. To verify that it is not a measure, it is sufficient to take , and note that . In other words. the countable additivity property is not satisfied. However, does satisfy the finite additivity property.

Complete Measures

Consider a measure space . A set is called a -null set (or simply null set) if . A property holds -almost everywhere (or simply almost everywhere) if satisfies and .

A measure space is called complete if contains all subsets of its null sets. An incomplete measure space can be constructed by taking and with . The set is a null set in this case, but .

Given an incomplete measure , the following theorem guarantees that a complete measure space this measure space can be extended to a complete measure space . The measure is called the completion of , and is called the completion of with respect to .

Theorem Suppose that is a measure space. Let and . Then, is a -algebra, and there is a unique extension of to a complate measure on .

Borel Measures and Lebesgue Measures

A measure whose domain is the Borel -algebra is called a Borel measure on . The following theorem provides a method for constructing Borel measures.

Theorem If is any increasing, right continuous function, there is a unique Borel measure on such that , for all . If is another such function, we have if and only if is constant. Conversely, if is a Borel measure on that is finite on all bounded Borel sets and we define , then is increasing and right continuous, and .

A few things should be noted regarding the previous theorem. The intervals can be replaced by intervals of the form ; in this case, the function would have to be left continuous. Additionally, the completion of , , is known as the Lebesgue-Stieljes measure associated to ; this complete measure has a domain that is strictly greater than the . Finally, taking gives rise to the Lebesgue measure.

References

  1. Folland, Gerald B. (1999). Real Analysis: Modern Techniques and Their Applications, John Wiley and Sons, ISBN 0471317160, Second edition.
  2. Craig, Katy. MATH 201A Lectures 4-5, 7-8. UC Santa Barbara, Fall 2020.