Serlo: EN: Continuity of volumes on rings

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In this article we derive the definition of continuity of volumes. We investigate how the notions of continuity from below and from above are related and learn about $\sigma$-rings as a domain of definition of continuous volumes.

Motivation for continuity of volumes

In the last chapter, we learned about volumes defined on rings defined as an abstract form of measuring (extensive) quantities. When measuring such quantities, we expect that small changes in the measured object will result in only small changes in the measurement result. Examples are

• Ingredients for cooking: if you add only a little more, weight and taste will only change slightly.
• Counting of objects: If one determines the number of an at most countably infinite quantity of objects, then this changes only little, if only few objects are added or taken away.
• The area/circumference of a circle: If the radius is changed only little, area and circumference also only change little.

A similar behaviour occurs for functins and is called continuity, there. So we would also like to define a continuity for sets - more precisely, sequences of sets

In fact, it is difficult to find extensive quantities in nature which are not continuous. It is also a natural and important question whether content functions are continuous (this should often be the case). Moreover, continuity has a very useful consequence: it allows an approximation of quantities to be measured. If small differences between sets cause only small differences between the measured values, then the error of the approximation can be controlled by the accuracy of the approximating sets. Thus, even "complicated" sets can be measured by approximating them with easier sets. An example is the approximation of the area of a circle (area complicated to describe) by rectangular figures (area easy to describe).

Exhaustion of an ellipse by rectangles

Continuity therefore seems to be a desirable property of a volume. Before we examine this notion in more detail: Is there any (intuitively) discontinuous volume at all? If yes, we know what to exclude from the definition.

Mathe für Nicht-Freaks: Vorlage:Beispiel

So definitely not every volumes is continuous,and it makes sense to define a precise notion of continuity. How can we formalize the continuity of a content mathematically? Following the sequence definition of continuity for real functions, we try the following definition:

But here we have to be careful: What is meant by $A_n\stackrel{n\to \mathrm{\infty }}{⟶}A$ if the $A_n$ are sets? We first need a notion of convergence for sequences of sets.

Sequences of sets

Imagine a sequence of intervals of length 1, which constantly move to the right by step length 1, i.e., $(A_n{)}_n\subseteq 𝒫(ℝ)$, $A_n=[n,n+1]$ . It is hard to determine a limit set, to which this sequence of sets converges. In contrast, consider the sequence of sets $A_n=[0,1+1/n]$. Those seem to shrink to a set $[0,1]$: Since the $A_n$ are contained within each other, one can take the set sequence to be an approximation of the set $[0,1]$ "from outside/above". Similarly, one can take increasing sequences of sets to be approximations of a set "from inside/below" (for example, the sequence of $A_n=[0,1-1/n]$ exhausting $[0,1)$). It is then meaningful to set the intersection or union of $A_n$ as the limit of the sequence.

• 
  The set sequence $A_n=[n,n+1]$ "escapes to the right" and hence does not converge.
• 
  The set sequence $A_n=[0,1+1/n]$ shrinks down to $[0,1]$ (blue region) and hence converges.

Let's have a look at some examples:

How do limits of monotonic set sequences get along with set rings? Is the limit in the ring? Let's look at the ring $ℛ$ of cuboids in ${ℝ}^2$ (i.e., rectangles) and two examples of monotonic growing set sequences in this ring. In the left picture a rectangle is approximated by a sequence of smaller rectangles. The limit is itself a rectangles and again lies in $ℛ$. On the right we see how a circle is approximated by rectangles. But the limiting set is no longer a rectangle and hence not in $ℛ$.

Obviously, the limit of a monotonic sequence of sets from a ring does not necessarily have to lie in the ring again. Our reasoning for this was rather intuitive with the rectangles (= 2D-cuboids), but we can also give a very concrete (not only intuitive) and short example:

Continuity from below and above

Definition of continuity

Equipped with this definition of limits of set sequences we can now make another attempt to define the continuity of a volume $\mu$ on a ring $ℛ$. We have just seen that for a monotonic set sequence on a ring $ℛ$ its limit is not necessarily again in the ring. Therefore, we must impose the restrictive condition that the limit of the set sequence lies again in the ring for the definition to make sense.

Let's take an example to see if this improved definition already describes the concept of continuity to our satisfaction.

So there are volumes which are intuitively continuous, but discontinuous in the sense of our definition. Obviously there is a problem with the definition of the continuity of above. Note: With the second definition of continuity, such "unreasonable" cases can also occur. For instance, the monotonic sequence of $\mu (A_n)$ can be bounded, while $\mu (A)=\mathrm{\infty }$ holds for the limit of the sequence of sets. However, calling such a volume discontinuous does not contradict our intuition. We saw an example of this at the beginning of the article.

Now what is the problem with the above definition of continuity? It is apparently due to the occurrence of the value infinity: while every $A_n$ is slightly "less" infinite than its predecessors, nevertheless $\mu (A_n)=\mathrm{\infty }$. We have to exclude this case in the definition of continuity. For this we require finiteness of the decreasing sequence, at least after passing a certain minimal index (like a sequence, which decreases starting from some minimal index).

Continuous volumes are a special kind of volumes.

Note: It is not necessary to require finiteness starting from an index within the continuity from below: If $\mu (A_N)=\mathrm{\infty }$ holds for an index $N\in \mathbb{N}$, then we have that $\mu (A_n)=\mathrm{\infty }$ also for all subsequent $n\ge N$ because of the monotonicity of the volume $\mu$. For the same reason, infiniteness then also holds for the limit $\bigcup _{n\in \mathbb{N}}{A}_n$.

Continuity from below implies continuity from above

continuity from above and continuity from below do not seem to be quite equivalent: At least, with the continuity from above one has to make restrictions which are not necessary with the continuity from below. How exactly are the two notions related? Does one imply the other? To get an answer, we try to find an analogy to sequences of real numbers.

Let $(x_n{)}_{n\in \mathbb{N}}\subseteq ℝ$ be a monotonically decreasing sequence of non-negative real numbers. You can make it a monotonically increasing sequence of non-negative real numbers by considering the sequence $(x_1-x_n{)}_{n\in \mathbb{N}}$. If this sequence converges to a value $y\in ℝ$, we can conclude that the original sequence $(x_n{)}_{n\in \mathbb{N}}$ converges to $x_1-y$.

Suppose we now know that a volume is continuous from below. Then we can infer the continuity from above by turning in the same way from a monotonically decreasing set sequence $(A_n{)}_{n\in \mathbb{N}}$ into a monotonically increasing set sequence $(A_1\setminus A_n{)}_{n\in \mathbb{N}}$. After that transformation, we can then exploit the continuity from below. Crucially, we must only need to consider decreasing sequences of sets with finite volume, so we don't get any problems with subtraction.

Continuity from above does not imply continuity from below

In the same way it should work if we know that a volume is continuous from above: we turn an increasing sequence into a descending one and exploit the continuity from above. But here we have to be careful! The continuity from above only holds for sequences which have finite content starting from an index. This condition is not guaranteed to be satisfied if we construct a descending sequence starting from an arbitrary increasing one. This problem can also occur with real-valued sequences: From a monotonically increasing sequence $(x_n{)}_{n\in \mathbb{N}}$ convergent to some $x\in ℝ$, we can construct the monotonically decreasing sequence $(x-x_n{)}_{n\in \mathbb{N}}$. But if $(x_n{)}_{n\in \mathbb{N}}$ is not upper bounded but tends to infinity, this is no longer possible.

For set sequences, on one hand, for every monotonically growing sequence $(A_n{)}_{n\to \mathrm{\infty }}$ with limit $A$ the sequence $(A\setminus A_n{)}_{n\in \mathbb{N}}$ is monotonically decreasing. But on the other hand, it can occur that $\mu (A\setminus A_n)=\mathrm{\infty }$ for all $n$. With such a sequence, we cannot use the continuity from above. So we cannot expect that the continuity from above always follows from the continuity from below. This is illustrated by the following example, which we already got to know in the first section as an example of a discontinuous volume:

We capture this observation:

For finite rings, continuity from above and below are equivalent

Intuitively, the continuity from above is weaker, because one has to impose the condition of finiteness of the volume starting from an index. So one "loses" some sequences if the volume does not take only finite values. In fact, "from above" and "from below" are equivalent for finite volumes:

Equivalent characterization of continuity from above

Finally, we give a simpler characterization of the continuity from above. It can be useful to prove the continuity of finite contents:

Examples for continuous volumes

Now that we know so much about continuous volumes, we can look at some concrete examples.

Mathe für Nicht-Freaks: Vorlage:Beispiel

Furthermore, one can easily show that finite linear combinations of continuous volumes are continuous again.

Sigma-rings

• 
  A ring (of sets) does not necessarily have to include all limiting sets.
• 
  However, a $\sigma$-Ring must also include all limiting sets.

We now know what continuity of a volume on a ring means. In the introduction we stated that continuity allows to measure sets by approximation, since small deviations of the sets induce only small deviations of the measured values. Thus, if a volume on a ring $ℛ$ is continuous, it should be possible to measure not only the sets from $ℛ$, but also all with sets from $ℛ$ that can be approximated. The approximable sets are even the limits of monotonically increasing or decreasing sequences of sets. So we have that for continuous volumes it makes sense to use a ring as domain of definition, which also contains the limit values of such sequences:

A $\sigma$-ring is a special ring. The arrow means "is included in"

For the continuity of a volume the continuity of below was sufficient, because one can construct an increasing set sequence out of every decreasing one. In the same way it is sufficient to formulate the closedness of $ℛ$ only for limits of increasing set sequences: If $(A_n{)}_{n\in \mathbb{N}}$ is a monotonically decreasing set sequence with limit $A$, then the sequence of $B_n=A_1\setminus A_n$ is a monotonically increasing sequence with limit $A_1\setminus A$. Since $ℛ$ is a ring, so in particular stable under differences, we have that Vorlage:Einrücken for all $n$ and Vorlage:Einrücken For the equalities on the left hand side we have exploited $A_n\subseteq A_1$ and $A\subseteq A_1$ respectively. So it is enough to require that limit values of monotonically increasing set sequences are again in the set system $ℛ$, and we have the equivalent definition:

In the literature $\sigma$-rings are often defined differently. We prove that the following is an equivalent characterization.

Examples for sigma-rings

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