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In the following articles on measure theory, we will gradually introduce various mathematical objects. These articles tell a story in which we will follow possible considerations of a mathematician, so we introduce new objects only when we really need them.
This article summarizes the objects in a compact way, so you can easily compare them.
The basis of measure theory is always a large basic set , for which we want to assign a measure to subsets that is, a number that indicates how large is. In many cases, however, not every subset is suitable for such an assignment. For instance the Banach-Tarski paradox or the Vitali sets show this.
Those sets to which, we can assign a suitable measure will simply be called measurable. We put them into a set system. So the is a set containing sets itself (like a bag in which there are even more bags), e.g. with .
To do computations with measures (i.e., addition, subtraction), we would like to perform operations with the sets, like unions , intersections or taking complements . And this without "getting kicked out of ", if possible. In mathematics, one therefore classifies set systems into different types, depending on how many operations we can perform without getting kickes out of and on other nice properties which they may satisfy:
The -algebra is the most special and most frequently encountered set system type. Here we can "afford relatively many operations" which avoids problems of the kind " is not defined on this set". is an-algebra if and only if
If a sequence of sets is in , then also
An algebra (of sets) also satisfies these 3 axioms, but the 3rd is only required to hold for finite sequences . The "" here stands for a countably infinite union of sets. If one omits it, then "only" finite sequences are allowed and one gets a more general set system. That is, there are "more algebras than -algebras". A set system is an algebra if and only if
If a sequence of sets is in , then also
A -ring (also denoted ) satisfies all conditions of the -algebra, except for 1. That is, we also allow set systems containing only smaller sets. E.g., there could be a maximal set containing all . A system is a ring if and only if
Immer wenn eine Folge aus Mengen in liegt, dann ist auch
Sometimes it is required as an additional condition that is not empty, i.e. . As soon as a ring contains any set , it always contains also the empty set .
A ring (of sets) is obtained equivalently by taking away from the -ring the -property. That means, we allow only finite in our condition. From the axioms of the -algebra only the 2nd and the 3rd in "finite" form are valid:
If a sequence of sets is in , then also
* The Dynkin-system is a separate type of set system. We will need it later to describe when measures match. The 3 axioms are:
For every pair of sets with we have
For countably many, pairwise disjoint sets we have .
Further set systems, that do not appear in the articles, are:
The monotone class: a type of set system containing all limits of monotonically increasing or decreasing set sequences . That means,
If form a monotonically increasing sequence, i.e., , then we have
If form a monotonically decreasing sequence, i.e., , then we have
The semi-ring is a generalization of the ring. The essential point is that no longer lies in , but only needs to be represented by a disjoint union of sets from it. The condition (which always holds on rings) must thus be required separately. Instead of union stability one also demands cut stability:
On the set systems defined above, we now try to define functions (or ) that intuitively measure the "size" of a set. The intuition "measure a size" can be translated into several desirable properties. For example, an empty set should have size 0, so . The more of these properties hold, the more the function matches our intuition to measure the size of sets.
Depending on how many and which of these desirable properties are satisfied by , we divide these functions into different classes. The most specific class is the measure, which has relatively many good properties and is therefore often used in mathematics, e.g., with in probability theory and statistics.
A measure is a function on a -algebra with the quite intuitive property that the empty set has measure 0 and when joining sets that do not overlap, their measures must also be added:
is -additive, i.e.,
A pre-measure is in principle the same as a measure, but needs to be defined only on a -ring . So the set can be in , but doesn't need to be. Thus it holds that
is -additive, i.e.,
A volume on a ring is a kind of "measure without -property for additivity". So we require the additivity only for finite unions:
The following two classes of functions are not additive, but only sub-additive and therefore get their own letter . I.e., if one unites, for example, a set with and one with there could be (or we could have any number )! This contradicts our intuition of "measuring a size". Therefore we give the functions a separate symbol .
The outer volume on the power set is defined in analogy to the volume above. But instead of additivity, one only requires sub-additivity:
is sub-additive, so from we get
An outer measure on the power set is the -version of the outer volume. That is, subadditivity is required even for countably infinite unions instead of finite unions. Thus the sub-additivity becomes a -sub-additivity, where the stands for "countably infinitely many".
is -sub-additive, so from we get
Examples: separating set system definitions[Bearbeiten]
The definitions of the set systems above are quite abstract and it is not obvious why some of them might not be equivalent. In the following examples, we will see how to separate the set system types
That means, we find examples that are of one but not a second type, respectively. In addition, you may find some (out of many possible) visualizations for those abstract definitions.
Example (Rings vs. -rings)
We construct a set system in which "limiting sets" are not contained: Let be a sequence of real numbers. These are elements of the interval . We choose the basic set to be slightly larger (this is perfectly allowed) and define the set system
i.e., all half-open intervals with endpoints from the sequence. This is not yet a ring. But we can make it a ring:
Let be the set system of all finite unions and intersections of intervals from (which are again finite unions of half-open intervals with as endpoints). The system is a ring (the "ring generated by "), but is not a -ring: the limiting sets
are not included. However, we can turn into a -ring by including the "limiting sets" for all . The set system
is again a -ring (and thus also a ring).
The system is "only" a ring (of sets).
The system is at the same time a -ring.
Example (-algebras vs. -rings)
We copy the setting from the previous example: The set system consists of all intervals with endpoints in the sequence . We choose the basis set to be (see figure).
The set system is a -algebra.
The set system is "only" a -ring.
The set system generates a ring (by finite unions and intersections). Joining it with the limit sets yields a -ring
However, the set system is not a -algebra, since it does not contain the basic set (likewise the empty set is not contained). In a sense, the set system is "too small". We can make it a -algebra: For this we add and the empty set , as well as the complement sets for all . From these sets we again form all arbitrary unions us intersections. and add them to the set system.
The result is a -algebra consisting of arbitrary unions of half-open intervals, with endpoints in .
The only difference to is that 2 is also allowed as endpoint and the empty set is included. -algebras can hence thought to be larger than mere -rings.
We can make even larger by adding a finite set of arbitrary points to the set of allowed endpoints. From this we can construct a "larger algebra" by choosing as a set system of arbitrary unions of half-open intervals between the endpoints (which is shown in the figure above).
Example (-Algebren vs. Algebren)
We use again the setting of the two examples above: On the interval we define a sequence of endpoints .
The set system , which consists of arbitrary finite unions of intervals of the form is a ring of sets. However, it is not a ring.
If we add 1 to the possible endpoints (that is, are allowed as endpoints of half-open intervals), and also allow countably infinite unions, we obtain a -ring. However, it is not a -algebra, since it does not contain .
If we add an additional 2 to the possible endpoints, we get a -algebra. Now, if we add finitely many more arbitrary endpoints, we still end up with -algebras, which are, however, larger, and will be denoted by .
The -algebras and can be very easily transformed back into an algebra by taking out the endpoint 1.
We define the set system to contain any countable unions of intervals with the same endpoints as those from , except 1. Then is an algebra, because contains . But taking it out destroys the -property. That is, a countable union of sets (which contains, say, 1 as an endpoint) need no longer be in .
Similarly, we can transform the -algebra into an algebra by removing the endpoint 1. The -property will then no longer hold.
The system is a -algebra.
If the endpoint 1 is taken away, "only an algebra" remains.
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