In this article we learn about -additivity of volumes and see how it can be used to characterize the continuity of volumes on rings. We call a volume with this property a pre-measure and thus define a notion central to measure theory: measures on -algebras.
In the previous article we learned about continuous volumes. Intuitively, we took a volume to be continuous if it allows the volume of a set to be measured by approximation. Based on this reasoning, we came up with a formal definition for the continuity of a volume. The following simpler formulation is equivalent to this, as we have seen:
Definition (Continuous volume)
A volume on a ring is called continuous if for every increasing set sequence with limit it holds that:
Approximation of simple measurable sets
The advantage of continuity is that one can determine the volume of a complicated set by approximation with sets that are easier to measure. But in order to be able to measure quantities by approximation, one must first know whether the volume is really continuous. And because we have defined continuity above exactly by this approximation property, we would first have to check for all set sequences whether the volumes of the sets really approximate the volume of the limit. Which is what we wanted to use. So we are essentially going around in circles
It would be nice to have a different characterization of continuity. Perhaps we can find one that resembles additivity, which is present for volumes anyway.
Definition of -additivity[Bearbeiten]
In the following, let be a volume on a ring . We know that for pairwise disjoint sets , by additivity we have that
Suppose is continuous. An infinite series is simply a limit of a sequence of finite sums, and we guess how additivity can be generalized for continuous volumes: Let be a sequence of pairwise disjoint sets in such that their union also lies in the ring . Then the sets form an increasing set sequence with limit . From the assumption that is continuous, it follows that
For a continuous volume the additivity is valid also for unions of countably infinitely many disjoint sets. This is of course subject to the union of the infinitely many disjoint sets being again in the domain of definition of . Volumes satisfying this property are called -additive, i.e. "countably additive":
Definition (-additive volume on a ring)
Let be a ring and a sequence of pairwise disjoint sets whose union lies also within the ring . A volume is called -additive if
It is not excluded that the series on the right side of the equation diverges, i.e. has the value infinity.
Characterization of continuity (on rings)[Bearbeiten]
We have seen that continuous volumes on rings are -additive. Let us recall our original goal: to find an alternative characterization of continuity. We want to investigate whether -additivity is suitable as such a characterization.
So now let be a -additive volume on a ring Let further be a monotonically increasing set sequence whose limit is again in . Let us try to prove the continuity of , i.e., the property
In order to exploit the -additivity, we need to transform the sequence of (not necessarily pairwise disjoint) into a sequence of pairwise disjoint sets whose union is also equal to . To do this, we take each of the sequence and cut out the part already contained in the previous sequence members: define the sets
Since rings are stable under taking set differences, the sequence of pairwise disjoint is also in . Further, and hence holds. So we have that
where in we have exploited the assumption that is a -additive volume.
Overall, our considerations show that for volumes on rings continuity and -additivity are equivalent. We have thus found an alternative characterization of continuity closely related to additivity:
Theorem (Equivalence of continuity and -additivity on rings)
For a volume on a ring , the following two statements are equivalent:
- is continuous,
- is -additive.
Proof (Equivalence of continuity and -additivity on rings)
: Let be continuous and let be a sequence of pairwise disjoint sets with . The sets lie in and form a monotonically growing set sequence. With the continuity of , we get
: Let be -additive and let be a monotonically growing set sequence with limit . Define for and . We have that and therefore also . It follows, using the additivity of , that
So is continuous.
Sometimes volumes are considered on domains of definition other than rings (such as so-called semi-rings). But for the equivalence of continuity and -additivity it is important that the volume is really defined on a ring: In the proof it is needed that the domain of definition is closed under set differences and finite unions.
Wir erinnern zunächst an ein Beispiel aus dem Artikel über stetige Inhalte.
Dort betrachten wir die Grundmenge und den Inhalt , der von einer beliebigen Teilmenge der natürlichen Zahlen bestimmt, ob sie endlich oder unendlich ist:
Der Inhalt wurde als unstetig erkannt, da die Bedingung der Stetigkeit für die aufsteigende Mengenfolge der Mengen mit Grenzwert nicht erfüllt ist. Tatsächlich ist er auch nicht -additiv. Ein Gegenbeispiel sind die paarweise disjunkten Mengen , die man wie oben durch Bilden der Differenzen aus den gewinnen kann. Für diese gilt
Ein Beispiel für einen -additiven (und also stetigen) Inhalt auf einem Ring ist dagegen der Inhalt mit , ebenfalls auf der Potenzmenge definiert, der die Anzahl der Elemente einer Teilmenge von bestimmt. (Dieser wurde hier genauer behandelt.) Es ist offenkundig, dass dieser Inhalt -additiv ist: Sind paarweise disjunkt, so gilt natürlich
Genauso ist natürlich jeder stetige Inhalt -additiv, wie unsere Überlegungen im vorherigen Abschnitt gezeigt haben. Beispiele für stetige Inhalte haben wir im Artikel zu stetigen Inhalten gesehen.
For volumes satisfying the useful -additivity, we will use a separate term:
A pre-measure is a
-additive and hence special continuous volume
A function on sets on a -ring is called a premeasure if for all sequences of pairwise disjoint sets in it holds that:
One can also define the notion of a pre-measure in general on rings, then one simply requires that is a -additive volume. That is one requires only for sequences of pairwise disjoint sets whose union is contained in (on -rings this is always the case).
Every pre-measure is also a volume. The non-negativity as well as holds by definition, the finite additivity we get from the -additivity by choosing all starting from a certain index.
For volumes, as shown in the sigma-additivity section, the equivalence between continuity and -additivity holds. Because -additive volumes are even pre-measures, a volume is continuous if and only if it is a pre-measure.
Sigma-algebras and measures[Bearbeiten]
We defined what a pre-measure is and thus characterized (on rings) continuity of volumes alternatively. As a natural domain of definition of a continuous volume we had learned -rings, since they are rings which additionally contain the limits of monotone set sequences.
A -algebra must contain the basic set
Recall: A -ring might also be a "smaller system of sets", not including .
Let be a -ring. It is useful to require that the basic set be "measurable", i.e., .
This is important, for example, in probability theory, where is the certain event. Moreover, with we obtain directly the complement stability via the difference stability of rings, which is often useful (e.g. counter-events in probability theory).
-algebra is a very special set system
A -ring with is called -algebra. That means, a -algebra must satisfy
- : sequence of sets in
So basically, a -algebra is a kind of "larger version of a -ring", as a -algebra must always contain the larges possible set , while a -ring not necessarily has to.
There is another common and equivalent definition of a -algebra, which is often easier to verify in practice.
A set system with
- : sequence of sets in
is called a -algebra.
Theorem (Both definitions of a -algebra are equivalent)
The definitions agree in two out of three points. We now show the equivalence of the third points.
- Let be a -algebra according to the first definition. Then by complement stability and by we have for all that . Thus is a -algebra in the sense of the second definition.
- Let be a -algebra according to the second definition. Let . Then, because of complement stability and union stability, . Thus is also a -algebra in the sense of the first definition.
So the definitions are equivalent.
Definition: measure, measurable space, measure space[Bearbeiten]
The crucial property of a pre-measure is its additivity with respect to countable disjoint unions as long as the countable disjoint union is again contained in the ring. For -algebras this is always the case. So we don't have to check the property, which makes our life a lot easier. hence, pre-measures on -algebras are easy to handle and appear a lot in mathematics. They have an own name: "measures".
The notion of a "measure" is actually the dominant one in mathematics. "Pre-measure" can rather be seen as generalizations of it, which are defined on "smaller" rings or -rings and are then extended to "larger" -algebras.
A measure is a special kind of pre-measure - and at the same time that function on sets with the most desirable properties.
A pre-measure is called measure if is a -algebra.
Definition (measurable space and measure space)
If is a -algebra with basic set , we call a measurable space.
Moreover, if is a measure on , we call a measure space.
A special case of measures are the so-called probability measures. For a set of elementary results , they assign to each event the probability that an outcome of a random experiment lies in . In this notion, the certain event () should have probability . So we say that a measure is a probability measure, if and only if :
Definition (Probability space and probability measure)
A measure space with is called a probability space. is then called probability measure. The elements of the -algebra are called events.
At this point, we see why it is crucial to consider measures in Probability theory and not just pre-measures: A pre-measure is only defined on a ring or a -ring, and this ring does not contain . So the statement might make no sense, if we are given only a "pre-measure"!
Examples for measures[Bearbeiten]
We now consider a few examples of measures on -algebras.
The first three examples are more or less trivial. Here, let be an arbitrary basic set and be a -algebra over .
Example (Zero measure)
Let with for all . Then is obviously a measure and is a measure space. We call the zero measure.
Another trivial example is given by
Again, it is clear that this is a measure: By definition, . Let be a sequence of disjoint sets. Then : If holds for each , then is on the left and a sum of zeros is on the right, so equality holds. If at least one of the is nonempty, then stands on both sides of the equation.
The next example can also be considered for any basic set , but is only of interest if it is overcountable.
Let be defined by
It is easy to see, as in the previous example, that is a measure: obviously holds. Let be a sequence of disjoint sets. Then : If is countable for every , then as a countable union of countable sets is itself countable. Consequently, is on the left and a sum of zeros is on the right, so equality holds. If at least one of the is overcountable, then we have on both sides of the equation.
The following examples are a bit less trivial:
Example (A volume on a finite basic set always generates a measure space)
Let be a volume over a finite basic set defined on the whole power set . Then is a measure space: It is clear that is a -algebra. We show that a volume over a finite basic set is already a measure. What we have to prove for this is -additivity: let be a sequence of pairwise disjoint sets in . From the finiteness of it follows that for all but finitely many .
Let be for all . Then . Thus is a -additive volume on a -algebra, that is, a measure.
Example (Counting-measure over )
Consider the measurable space and , . Then is a measure.
Obviously, . What remains to be shown is the -additivity. Let be a sequence of pairwise disjoint sets in . We distinguish two cases:
- We finitely often have . In that case it suffices to see that is a volume on , as shown in this example.
- We infinitely often have . In that case, it follows from the disjointness of that holds. Therefore, we have , so .
Thus the additivity holds in both cases, so is indeed a measure.
Example (Dirac measure)
Let be a measurable space, . Then is called the Dirac measure. It is easy to see that is a measure. Obviously . Now, let be any sequence of pairwise disjoint sets in , then there are two possibilities.
- lies in exactly one set . In that case .
- for all . Then
Thus the additivity holds and is indeed a measure.
We will get to know more interesting examples when we have taken a closer look at the construction of measures. For now, we don't even know if there is a -algebra over containing the intervals and on which the elementary geometric length is a measure.