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.
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:
- Let be a disjoint sequence of sets such that each . Then, .
If the previous conditions are satisfied, the structure is called a measure space.
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 .
Let be a measure space.
- Finite Additivity: Let be a finite disjoint sequence of sets such that each . Then, . This follows directly from the defintion of measures by taking .
- Monotonicity: Let such that . Then, .
- Subadditivity: Let . Then, .
- Continuity from Below: Let such that . Then, .
- Continuity from Above: Let such that and for some . Then, .
- 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:
- The measure is semifinite if and only if for every .
- 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:
- When fixing , the resulting measure is referred to as the counting measure.
- 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.
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.
- 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.