Serlo: EN: Volumes on rings

{{#invoke:Mathe für Nicht-Freaks/Seite|oben

|info=Mathe für Nicht-Freaks: Vorlage:Banner/Maßtheorie Autorenwerbung

}}

Extensive quantities

"Measuring" in the sense of measure theory means quantify: What is asked is not "what value has ...?", "how fast/hot/bright is ...?", but "how much of ...?", "how big/heavy/numerous is ...?". So we are interested in quantities that change with the size of the underlying system: doubling the system doubles the quantity. In physics, quantities with this property are called extensive quantities. An example is mass: if we join two systems of boxes $K_1$ and $K_2$ with mass $m_1$and $m_2$, the resulting system $K$ has the larger mass $m_1+m_2$. By contrast, quantities that do not change with the size of the underlying system are also called intensive. For example, temperature is an intensive quantity: pouring two liquids of one temperature together does result in the same temperature, and not the sum of the temperatures.

So whenever we quantify something in measure theory, we are measuring an extensive quantity. Other examples of such quantities are: Mathe für Nicht-Freaks: Vorlage:Beispiel So we can understand the measure theory as a mathematical theory of measuring extensive quantities.

Volume-measuring functions

How can we grasp the measuring of such quantities mathematically? Obviously, by measuring a quantity, an assignment is described: A certain object (geometric body, event, accumulation of things) gets assigned exactly one "volume". Thus, the measuring of a quantity can be described by a function. The domain of definition of this function contains the objects to be measured. These can be understood as subsets of a huge set $\mathrm{\Omega }$. For example, (physical) objects. such as cubes, cuboids, spheres, cones, etc. can be seen as subsets of ${ℝ}^3$:

Likewise, one can consider subsets of $\mathbb{N}$ when measuring an extensive quantity counting objects. So the domain of definition will be a set system $𝒞\subseteq 𝒫(\mathrm{\Omega })$ of subsets of a large basic set $\mathrm{\Omega }$ (to be determined more precisely in each case) and the function will be a function on sets. The function values correspond to the volumes assigned to these sets and are scalars in $ℝ$. For now, let us call such a function a volume-measuring function.

From a mathematical point of view, a volume-measuring function is nothing more than a function on sets with values in $ℝ$. We investigate which further properties a volume-measuring function should have in order to be able to describe the measurement of an extensive quantity in a meaningful way.

Which properties shall a volume-measuring function have?

Non-negativity

It makes intuitive sense to require non-negativity of volume-measuring functions. After all, how should a negative volume be interpreted? Admittedly, there are situations in which one also allows negative numbers as dvalues of a volume-measuring function (signed measures). One example is the measurement of a total charge, which is composed of positive and negative parts. Sometimes the range of values is generalized even further, so that also complex numbers can appear as function values (complex measures) or certain linear mappings (spectral measures). However, all these cases arise as generalizations from the non-negative real situation, so we will restrict ourselves to the real case at the being. Moreover, infinity should also be allowed as a function value in certain cases: For instance, the geometric volume of ${ℝ}^n$ should be infinite. So we require: a volume-measuring function maps to $[0,\mathrm{\infty }]:={ℝ}_0^{+}\cup \{\mathrm{\infty }\}$.

Monotonicity

Volume-measuring functions are intended to formalize the measurement of extensive quantities. Extensive quantities are characterized by the fact that they increase with increasing size of the underlying system. For example, when determining the number of atoms in a sample of matter, one should count more atoms after adding a certain amount of the matter. This property should also be found in volume-measuring functions: enlarging a geometric object makes it gain more volume. Mathematically, this can be described by the term monotonicity of a function on sets:

We will demand a volume-measuring function to be monotonic: larger sets have larger volumes.

Sub-additivity

The sets $A_1,\dots ,A_7$ cover the rectangle $A$.

$B_1,B_2,B_3$ cover $A$.

The monotonicity of the volume-measuring function guarantees that the value for a given set $A\in 𝒞$ is always less than or equal to the value for any superset $B\in 𝒞$. But is this also true if the superset $B$ is a union of several sets? Intuitively, this should be the case: If a set $A\in 𝒞$ is covered by sets $A_1,A_2,\dots ,A_n\in 𝒞$, then the volume-measuring function evaluated on $A$ should in no case yield a value larger than for the sum of the contents of the covering sets.

Mathematically expressed: for a volume-measuring function $\mu$ and sets $A,A_1,\dots ,A_n\in 𝒞$ it should hold that:

In particular, this property should be satisfied if the union of the covering sets $A_1,\dots ,A_n$ itself is no longer in the domain of definition $𝒞$. But this property is not yet expressed by the monotonicity!

Thus, the property of functions on sets that monotonicity is preserved even in the case of finite covers has to be required additionally. We call it sub-additivity:

If the covering of the set $A$ consists of only one set $B$, this property corresponds exactly to monotonicity. So we can take sub-additivity as a generalization of monotonicity and replace the requirement of monotonicity of volume-measuring functions by this generalized version: measuring functions must be sub-additive.

Additivity

The properties of monotonicity or more generally of sub-additivity required so far make only an "approximate" statement: the function values of a sub-additive function on sets may be above the "real" value. As an example:

Disjoint union with $n=7$

Obviously, for a volume-measuring function to be "exact", this property must be additionally required: If a set $A\in 𝒞$ is exactly covered by finitely many, pairwise disjoint sets $A_1,\dots ,A_n$, then the function value of the volume-measuring function evaluated at $A$ is said to be equal to the sum of the function values for the individual $A_i$.

It is hence important whether the sets within a union are pairwise disjoint. To make explicit the pairwise disjointness within a union, we introduce a new notation:

The above formulated condition of "exactness" of a function on sets is called (finite) additivity: for an exact covering of a set, decomposition and reassembly does not change its volume:

We fix this desirable property of a volume-measuring function to be exact by demanding: volume-measuring functions must be additive.

It is important to note that due to this limiting condition, the additivity is even more dependent on the domain of definition $𝒞$ (i.e., a set system), than the sub-additivity. In particular, the additivity of a function on sets does not in general follow from its sub-additivity or monotonicity. A counter-example is a function on sets, whose domain of definition $𝒞$ does not contain disjoint sets:

However, we will see later that additivity implies sub-additivity if the domain of definition $𝒞$ has "a sufficiently good structure".

Last, the additivity of volume-measuring functions makes it desirable that $\mu (\mathrm{\varnothing })=0$ holds: Because of additivity, $\mu (\mathrm{\varnothing })=\mu (\mathrm{\varnothing }\uplus \mathrm{\varnothing })=\mu (\mathrm{\varnothing })+\mu (\mathrm{\varnothing })$ must hold, and this condition is only satisfiable for $\mu (\mathrm{\varnothing })=0$ or $\mu (\mathrm{\varnothing })=\mathrm{\infty }$. If now $\mu (\mathrm{\varnothing })=\mathrm{\infty }$, then due to monotonicity $\mu (A)=\mathrm{\infty }$ would have to hold for all sets $A\in 𝒞$. To exclude this pathological case, one additionally demands: For volume-measuring functions, $\mu (\mathrm{\varnothing })=0$ must hold.

Conclusion

Let us recap the properties that a volume-measuring function should have in order to meaningfully describe the measurement of an extensive quantity:

• Such a function should map to $[0,\mathrm{\infty }]$. In most cases, this makes sense. But we can still generalize to further extended ranges of values.
• It should be sub-additive. We derived sub-additivity as an improved form of monotonicity that is preserved even in the presence of finite covers. In particular, any sub-additive function on sets is also monotonic. The monotonicity itself reflects the characteristic property of extensive quantities to grow with increasing size of the system under consideration.
• It should be additive. Additivity corresponds to the property of a volume-measuring function to be "exact" and to be compatible with decomposing measured objects into finitely many parts and reassembling them.
• It should assign the value zero to the empty set. This makes intuitive sense and also serves to exclude the case that the volume-measuring function is constantly infinite.

Rings: the domain of definition

In the considerations about additivity a difficulty has already appeared: Whether a set function is additive also depends on its domain of definition. The more sets are contained in the definition range, the more set ocmbinations we have to check for additivity, so it gets more difficult. After all, how large can a system $𝒞$ containing subsets of $\mathrm{\Omega }$ be made(as the domain of definition of a volume-measuring function) without destroying its additivity? Indeed, the answer is not generally "the power set $𝒫(\mathrm{\Omega })$" (whichh would be the largest possible set system). We have to stop before! An example where this happens is the elementary geometric volume on the ${ℝ}^n$:

The volume problem

The goal is to define an additive set function that describes the elementary geometric volume in ${ℝ}^p$. The problem of defining this function on all of $𝒫({ℝ}^p)$ is also called the volmue problem. And it will turn out to be unsolvable!

However, Hausdorff, Banach and Tarski were able to prove that such a measure cannot be found:

• The volume problem is unsolvable for the ${ℝ}^p$ if $p\ge 3$. (Hausdorff, 1914)
• The volume problem is solvable for the ${ℝ}^1$ and the ${ℝ}^2$, but it is not uniquely solvable. (Banach, 1923)

The first case looks somewhat paradox: we would expect such a volume naturally to exist (why should subsets of the ${ℝ}^p$ for$p\ge 3$ do not have a volume?) However, this is not the case! Building on Hausdorff's result, Banach and Tarski showed the following theorem, which is also known as reflecting the "Banach-Tarski paradox":

This theorem makes vividly clear that the volume problem on the ${ℝ}^p$ cannot be solvable for $p\ge 3$. The implications would be absurd: it follows from the theorem that a sphere (think of a pea or an orange) can be suitably divided into finitely many parts and assembled into a sphere the size of the sun. (Of course, this does not work in real life, simply because physical bodies are not continuous sets of points, but are composed of atoms. The quantities of such a decomposition are extremely complex and can best be illustrated as "point clouds", which in general cannot be stated explicitly).

Mathe für Nicht-Freaks: Vorlage:Warnung

Rings of sets

So, in general, we cannot hope to define an additive set function on the whole power set. This makes it necessary to think more carefully about the set system $𝒞$, which is to serve as the domain of definition of volume-measuring functions. Mathematicians spent some considerable time on defining classes of such set systems and therefore, the literature on measure theory provides a whole zoo of such set systems: Semi-rings, rings, $\sigma$-rings (pronounced "sigma-rings"), algebras, $\sigma$-algebras, Dynkin systems, monotonous classes, ... You may find an overview in this article

Intuitively, the domain $𝒞$ of an additive (and subadditive) volume-measuring function should be stable under the following set operations: the disjoint union of finitely many sets "$\cup$" and its counterpart, forming differences of sets "$\setminus$". Moreover, if one has the difference operation available, one can make arbitrary unions "artificially" disjoint: if $A$ and $B$ are (not necessarily disjoint) sets, then $A\cup B=A\uplus (B\setminus A)$, where the two united sets on the right-hand side of the equation are again in $𝒞$. So one can forget about disjointness and speak of arbitrary finite unions. Finally, to exclude trivial cases, we require $𝒞\ne \mathrm{\varnothing }$ and can now define:

Examples for rings of sets

Over a base set $\mathrm{\Omega }$ there are always the two rings $ℛ=\{\mathrm{\varnothing }\}$ and $ℛ=\{\mathrm{\varnothing },\mathrm{\Omega }\}$.

This is already an example containing a lot of sets. Especially, for finite $𝒞$, the ring $ℛ$ can be chosen to be all of the power set $𝒫(𝒞)$. For base sets with infinitely many points, such as $𝒞=\mathbb{N}$, one can also easily construct rings:

Another important example of a ring in case $𝒞={ℝ}^n$ is the following:

Volumes on rings

We have now found some properties that a volume-measuring functions should satisfy in order to meaningfully describe the measurement of an extensive quantity: non-negativity, subadditivity (especially monotonicity) and additivity. As natural domains of definition of such functions we juts defined rings.

Additivity is sufficient on rings

Above, we noticed that from the additivity of a volume-measuring function, the subadditivity does not follow in general. And not even its monotonicity - at least not if nothing else is known about the domain of definition $𝒞$. But if the set system $𝒞$ is a ring, then the monotonicity and the subadditivity already follow from the additivity and non-negativity of the volume-measuring function. The reason is that by cut-stability of the set system, one can easily make unions "artificially" disjoint and thus exploit the property of additivity. When proving this statement, it is clever to first show the somewhat simpler property of monotonicity and then subadditivity.

Definition

Thus, one can characterize the monotonicity and subadditivity of a volume-measuring function defined on a ring by its non-negativity and additivity alone. So we can now summarize all the properties that a measurement function should possess in the following definition:

The condition $\mu (\mathrm{\varnothing })=0$ makes intuitive sense and also serves to exclude the pathological case $\mu \equiv \mathrm{\infty }$.

Examples

A particularly important example is the so-called elementary geometric volume:

Further properties of volumes

We collect properties of volumes. In the following, let $ℛ$ be a ring (in particular closed under differences and finite unions) and let $\mu :ℛ\to [0,\mathrm{\infty }]$ be a volume.

Another property of finite volumes (i.e. $\mu (A)<\mathrm{\infty }$ for all sets $A\in ℛ$) is the so-called inclusion-exclusion principle. It is important in probability theory (where $\mu (A)\le 1<\mathrm{\infty }$), and traces the volumes of a union of sets to a sum of volumes of intersections. Note that for a ring $ℛ$ and $A,B\in ℛ$, we also have $A\cap B=A\setminus (A\setminus B)\in ℛ$ . Finite intersections of sets from $ℛ$ thus lie again in $ℛ$.

{{#invoke:Mathe für Nicht-Freaks/Seite|unten}}