Serlo: EN: Pre-measures and measures

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In this article we learn about $\sigma$-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 $\sigma$-algebras.

Sigma-additivity

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:

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 $\sigma$-additivity

In the following, let $\mu$ be a volume on a ring $ℛ$. We know that for pairwise disjoint sets $B_1,\dots ,B_n\in ℛ$ , by additivity we have that

Suppose $\mu$ 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 $(B_n{)}_{n\in \mathbb{N}}\subseteq ℛ$ be a sequence of pairwise disjoint sets in $ℛ$ such that their union $A:=\underset{i=1}{\overset{\mathrm{\infty }}{⨄}}{B}_i$ also lies in the ring $ℛ$. Then the sets $A_n:=\underset{i=1}{\overset{n}{⨄}}{B}_i$ form an increasing set sequence with limit $A\in ℛ$. From the assumption that $\mu$ is continuous, it follows that

For a continuous volume $\mu$ 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 $\mu$. Volumes satisfying this property are called $\sigma$-additive, i.e. "countably additive":

Characterization of continuity (on rings)

We have seen that continuous volumes on rings are $\sigma$-additive. Let us recall our original goal: to find an alternative characterization of continuity. We want to investigate whether $\sigma$-additivity is suitable as such a characterization.

So now let $\mu$ be a $\sigma$-additive volume on a ring $ℛ.$ Let further $(A_n{)}_{n\in \mathbb{N}}\subseteq ℛ$ be a monotonically increasing set sequence whose limit $A:=\bigcup _{n=1}^{\mathrm{\infty }}{A}_n$ is again in $ℛ$. Let us try to prove the continuity of $\mu$, i.e., the property

In order to exploit the $\sigma$-additivity, we need to transform the sequence of (not necessarily pairwise disjoint) $A_n$ into a sequence of pairwise disjoint sets whose union is also equal to $A$. To do this, we take each $A_n$ 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 $B_n$ is also in $ℛ$. Further, $A_n=\underset{i=1}{\overset{n}{⨄}}{B}_i$ and hence $\underset{i=1}{\overset{\mathrm{\infty }}{⨄}}{B}_i=\bigcup _{i=1}^{\mathrm{\infty }}{A}_i=A$ holds. So we have that

where in $(*)$ we have exploited the assumption that $\mu$ is a $\sigma$-additive volume.

Overall, our considerations show that for volumes on rings continuity and $\sigma$-additivity are equivalent. We have thus found an alternative characterization of continuity closely related to additivity:

Examples

Wir erinnern zunächst an ein Beispiel aus dem Artikel über stetige Inhalte. Dort betrachten wir die Grundmenge $\mathbb{N}$ und den Inhalt $\mu :𝒫(\mathbb{N})\to [0,\mathrm{\infty }]$, 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 $A_n=\{1,\dots ,n\}$ mit Grenzwert $A=\mathbb{N}$ nicht erfüllt ist. Tatsächlich ist er auch nicht $\sigma$-additiv. Ein Gegenbeispiel sind die paarweise disjunkten Mengen $B_n=\{n\}$, die man wie oben durch Bilden der Differenzen $A_n\setminus A_{n-1}$ aus den $A_n$ gewinnen kann. Für diese gilt

Ein Beispiel für einen $\sigma$-additiven (und also stetigen) Inhalt auf einem Ring ist dagegen der Inhalt mit $\mu (A)=|A|$, ebenfalls auf der Potenzmenge $𝒫(\mathbb{N})$ definiert, der die Anzahl der Elemente einer Teilmenge von $\mathbb{N}$ bestimmt. (Dieser wurde hier genauer behandelt.) Es ist offenkundig, dass dieser Inhalt $\sigma$-additiv ist: Sind $B_1,B_2,\dots \subseteq \mathbb{N}$ paarweise disjunkt, so gilt natürlich $\left|\underset{n=1}{\overset{\mathrm{\infty }}{⨄}}{B}_n\right|=\sum _{n=1}^{\mathrm{\infty }}|B_n|.$

Genauso ist natürlich jeder stetige Inhalt $\sigma$-additiv, wie unsere Überlegungen im vorherigen Abschnitt gezeigt haben. Beispiele für stetige Inhalte haben wir im Artikel zu stetigen Inhalten gesehen.

Pre-measures

For volumes satisfying the useful $\sigma$-additivity, we will use a separate term:

A pre-measure is a $\sigma$-additive and hence special continuous volume

Every pre-measure is also a volume. The non-negativity as well as $\mu (\mathrm{\varnothing })=0$ holds by definition, the finite additivity we get from the $\sigma$-additivity by choosing all $A_n=\mathrm{\varnothing }$ starting from a certain index.

For volumes, as shown in the sigma-additivity section, the equivalence between continuity and $\sigma$-additivity holds. Because $\sigma$-additive volumes are even pre-measures, a volume is continuous if and only if it is a pre-measure.

Sigma-algebras and measures

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 $\sigma$-rings, since they are rings which additionally contain the limits of monotone set sequences.

Definition: $\sigma$-algebra

• 
  A $\sigma$-algebra must contain the basic set $\mathrm{\Omega }$
• 
  Recall: A $\sigma$-ring might also be a "smaller system of sets", not including $\mathrm{\Omega }$.

Let $ℛ$ be a $\sigma$-ring. It is useful to require that the basic set be "measurable", i.e., $\mathrm{\Omega }\in ℛ$. This is important, for example, in probability theory, where $\mathrm{\Omega }$ is the certain event. Moreover, with $\mathrm{\Omega }\in ℛ$ we obtain directly the complement stability via the difference stability of rings, which is often useful (e.g. counter-events in probability theory).

A $\sigma$-algebra is a very special set system

So basically, a $\sigma$-algebra is a kind of "larger version of a $\sigma$-ring", as a $\sigma$-algebra must always contain the larges possible set $\mathrm{\Omega }$, while a $\sigma$-ring not necessarily has to.

There is another common and equivalent definition of a $\sigma$-algebra, which is often easier to verify in practice.

Definition: measure, measurable space, measure space

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 $\sigma$-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 $\sigma$-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 $\sigma$-rings and are then extended to "larger" $\sigma$-algebras.

A measure is a special kind of pre-measure - and at the same time that function on sets with the most desirable properties.

Mathe für Nicht-Freaks: Vorlage:Definition A special case of measures are the so-called probability measures. For a set of elementary results $\mathrm{\Omega }$, they assign to each event $A\subseteq \mathrm{\Omega }$ the probability that an outcome of a random experiment lies in $A$. In this notion, the certain event ($x\in \mathrm{\Omega }$) should have probability $1$. So we say that a measure $\mu$ is a probability measure, if and only if $\mu (\mathrm{\Omega })=1$:

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 $\sigma$-ring, and this ring does not contain $\mathrm{\Omega }$. So the statement $\mu (\mathrm{\Omega })=1$ might make no sense, if we are given only a "pre-measure"!

Examples for measures

We now consider a few examples of measures on $\sigma$-algebras.

The first three examples are more or less trivial. Here, let $\mathrm{\Omega }$ be an arbitrary basic set and $𝒜$ be a $\sigma$-algebra over $\mathrm{\Omega }$.

The next example can also be considered for any basic set $\mathrm{\Omega }$, but is only of interest if it is overcountable.

The following examples are a bit less trivial:

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 $\sigma$-algebra over $ℝ$ containing the intervals $[a,b]$ and on which the elementary geometric length $\lambda ([a,b])=b-a$ is a measure.

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