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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 -rings as a domain of definition of continuous 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).
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.
Example (A discontinuous volume)
We consider the basic set and the volume , which assigns any subset of natural numbers either finite amount or infinity:
In the article volumes on rings we saw that is really a volume defined on the ring . Now consider for the sets . For these obviously approximate the basic set . But since every is finite, for every . So the function value is not approximated in the limit . So the volume is intuitively "discontinuous".
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:
Definition (Continuous volume, first attempt)
A volume on a ring is called continuous, if for a sequence with we have that .
But here we have to be careful: What is meant by if the are sets? We first need a notion of convergence for sequences of sets.
Imagine a sequence of intervals of length 1, which constantly move to the right by step length 1, i.e., , . It is hard to determine a limit set, to which this sequence of sets converges. In contrast, consider the sequence of sets . Those seem to shrink to a set : Since the are contained within each other, one can take the set sequence to be an approximation of the set "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 exhausting ). It is then meaningful to set the intersection or union of as the limit of the sequence.
The set sequence "escapes to the right" and hence does not converge.
The set sequence shrinks down to (blue region) and hence converges.
Definition (Monotonic sequence of sets)
Let be a set and a sequence of subsets.
The sequence is called increasing or monotonically increasing if for all . In this case we set
and write .
The sequence is called descending or monotonically descending if for all . In this case we set
and write .
With our definition we can speak of the limit of a set sequence only if it is monotonic. This is enough for us at this point. But one can define and for arbitrary set sequences, which always exist as for real number sequences. Then, we are able to define even more converging sequences of sets.
Let's have a look at some examples:
Example (Monotonic sequence of sets)
The sequence is monotonically decreasing with limit
The inclusion "" holds since for all . On the other hand, the limiting set cannot contain more elements, since none of the contains negative numbers and every number strictly greater than one is no longer in for an index , so it is not in the intersection either.
Another example for a monotonically decreasing set sequence is . One shows with similar arguments that for this sequence is the limit.
The sequence of is an example of a monotonically increasing set sequence. Its limit is
The inclusion "" holds since every lies in the set for some index . On the other hand, the limit cannot be a larger interval since none of the contains negative numbers or one.
Another monotonically increasing sequence of sets is . The limit is , as can be seen in a similar way.
How do limits of monotonic set sequences get along with set rings? Is the limit in the ring? Let's look at the ring of cuboids in (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 . 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 .
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:
Example (Limit is no element of the ring)
Consider the ring of all finite subsets of (we learned about this ring here). The sequence with is monotonically increasing and lies in . But its limit is all of and thus no longer lies in the ring of all finite subsets of .
Equipped with this definition of limits of set sequences we can now make another attempt to define the continuity of a volume on a ring . We have just seen that for a monotonic set sequence on a ring 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.
Definition (Continuous volume, second attempt)
A volume on a ring is called continuous from below (from above), if for every ascending (descending) set sequence with limit , it holds that
A volume is just called continuous if it is continuous of below and of above.
Let's take an example to see if this improved definition already describes the concept of continuity to our satisfaction.
Example (A problem with the definition)
In the introduction we learned about counting the elements contained in a set as an example of an (intuitively) continuous volume. Let us look at the set system
of all finite subsets of . In the article volumes on rings we saw that counting the elements of finite subsets of really defines a volume:
We now expect to be continuous from below and from above in the sense of the new definition. Let then be a monotonically growing sequence whose limit is also in . Since the converge to , the number of elements in must be the limit of the number of elements of . In other words, we have that
Thus is continuous of below in the sense of our definition. Let now be a monotonically decreasing sequence whose limit is also in . Again, for the same reasons, the continuity condition must be satisfied. Thus is continuous on the ring in the sense of the definition.
One can also consider the volume on the whole power set . Intuitively, counting of elements should be just as continuous on arbitrary subsets of as on finite subsets. In the argumentation, however, we should make a distinction between finite and infinite to be on the safe side:
Let again be a monotonically increasing sequence of subsets with limit . If contains only finitely many elements, then because of monotonicity, and we can argue as above. But since is the limit of , the argument works even if contains infinitely many elements: If , then the number of elements in the would not grow beyond a maximal index. Thus the sequence of would be constant from an index and in particular could not converge to an infinite set. So must hold and also for infinite , we have that
If the now form a monotonically decreasing sequence and all sets of the sequence are finite, then because of the monotonicity, also the limit is a finite set and we can argue as above. It works the same way if only finitely many of the are infinite: We can omit all infinite sequences elements without restriction, since omitting finitely many elements of a sequence does not change its limit, and get again a sequence of finite sets only. The situation is different if all of the sequence are infinite sets. To satisfy the continuity condition, we would need
meaning that the limit of the sequence contains infinitely many elements. But since the sequence is monotonically decreasing, this is not guaranteed. We have not excluded that a descending sequence of infinite sets converges to a finite set. Try to find such a sequence before reading on!
Consider the sequence with . The converge decreasingly to the empty set. But at the same time we have that for all . So
I.e., the volume , which intuitively should be continuous, is not continuous in the sense of our definition.
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 can be bounded, while 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 is slightly "less" infinite than its predecessors, nevertheless . 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).
Definition (Continuous volume (final definition))
A volume on a ring is called continuous from below if for every increasing set sequence with limit , it holds that
It is called continuous from above if for every decreasing set sequence with for one (and thus for all further) and with limit , it holds that
A volume is just called continuous if it is continuous from below and above.
The continuity of volumes or more generally of functions on sets is sometimes also called -continuity ("sigma-continuity"). The prefix is meant to remind of "sum" and means in this context something like "countable": -continuous volumes behave continuously at the transition from finite to countable unions or cuts.
Note: It is not necessary to require finiteness starting from an index within the continuity from below: If holds for an index , then we have that also for all subsequent because of the monotonicity of the volume . For the same reason, infiniteness then also holds for the limit .
Continuity from below implies continuity from above[Bearbeiten]
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 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 . If this sequence converges to a value , we can conclude that the original sequence converges to .
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 into a monotonically increasing set sequence . 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.
Theorem (continuity from below implies continuity from above)
Let be a volume on a ring . If is continuous from below, then is also continuous from above. In particular, continuity of a content from below is equivalent to just continuity.
Proof (continuity from below implies continuity from above)
Let be a monotonically decreasing set sequence with starting at an index . Since omitting finitely many sequences elements does not change the limit, we can assume without restriction. The limit lies in , as well. We want to show .
The sequence is monotonically increasing and lies in the ring , as well. Also the limit lies in . Furthermore, due to monotonicity, all these sets have finite content. From the continuity from below we know that holds. Since moreover for all , we have that . With the additivity of the volume it follows that
so . (Alternatively, one can argue directly with the subtractivity of volumes on rings). Likewise we have that . Since the contents of all involved volumes are finite, these differences make sense and by continuity from below,
Continuity from above does not imply continuity from below[Bearbeiten]
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 convergent to some , we can construct the monotonically decreasing sequence . But if is not upper bounded but tends to infinity, this is no longer possible.
For set sequences, on one hand, for every monotonically growing sequence with limit the sequence is monotonically decreasing. But on the other hand, it can occur that for all . 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:
Example (Continuity from above does not imply continuity from below)
The set sequence with is incraesing with . Thus the sequence of is a descending sequence with . While all are finite sets, the are all infinite. Let us now consider the volume that determines whether a subset of natural numbers is finite or infinite:
Obviously is continuous from above: The only descending set sequences that satisfy the finiteness condition are finite from an index, and then, their content is constantly .
But now the sequence of does not satisfy the finiteness condition from an index, so we cannot use it to infer the continuity from below of for the sequence of . And indeed,
So is not continuous from below. In particular, from the continuity of a content from above, we generally not get the continuity from below!
We already got to know this volume in the introduction as an example for a (intuitively) discontinuous volume. The example shows that is not continuous also in the sense of our definition.
We capture this observation:
From the continuity of a content from above, we can generally not imply the continuity from below.
For finite rings, continuity from above and below are equivalent[Bearbeiten]
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:
Theorem (For finite rings, continuity from above and below are equivalent)
Let be a finite volume on a ring , i.e.. for all . Then, we have
Proof (For finite rings, continuity from above and below are equivalent)
We have already shown that continuity from below always implies continuity from above. Consequently, it is now sufficient for the finite case to show the other direction.
So let be continuous from above and be a sequence converging from below to with sets from , i.e. .
We note that is always back in the ring . (stability under taking differences).
We construct a decreasing set sequence from the increasing one to exploit the continuity given above.
For this we define for all .
Since was increasing, is decreasing.
It follows that .
Thus is a set sequence converging to in . Because of the assumed continuity of above, we have
Here we also used the finiteness of in order to get continuity from above ( is always finite).
For every , due to additivity, .
Equivalent characterization of continuity from above[Bearbeiten]
Finally, we give a simpler characterization of the continuity from above. It can be useful to prove the continuity of finite contents:
Theorem (Continuity from above and in )
For a volume on a ring , the following statements are equivalent:
is continuous from above.
is continuous in , i.e. for all monotonically decreasing sequences with starting from some and we have that .
Proof (Continuity from above and in )
: This is simply the definition of "continuity of above" applied to .
: Let now be continuous in , and let with starting from some and .
Then is monotonically decreasing, and with the monotonicity property of volumes we have that starting from some .
Since moreover we have that , it follows that converges to from above.
By our assumption then .
Since is finite for sufficiently large , and since always holds, it follows that .
A ring (of sets) does not necessarily have to include all limiting sets.
However, a -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 is continuous, it should be possible to measure not only the sets from , but also all with sets from 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 ring over some basic set is called -Ring, if:
limits of increasing sets sequences in lie in ,
limits of decreasing sets sequences in lie 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 only for limits of increasing set sequences: If is a monotonically decreasing set sequence with limit , then the sequence of is a monotonically increasing sequence with limit . Since is a ring, so in particular stable under differences, we have that
for all and
For the equalities on the left hand side we have exploited and respectively. So it is enough to require that limit values of monotonically increasing set sequences are again in the set system , and we have the equivalent definition:
Definition (-ring (equivalent definition))
A ring over a basic set is called a -ring if for every increasing set sequence in its limit is also in .
In the literature -rings are often defined differently. We prove that the following is an equivalent characterization.
Theorem (Alternative characterization of -rings)
A set system over a basic set is a -ring exactly if:
Proof (Alternative characterization of -rings)
Let be a -ring. Since is a ring, properties 1 and 3 of this theorem are satisfied. Now let be a sequence in . As is a ring, finite unions of the form . Since is a -ring, and because is an increasing set sequence in , it follows that . So property 2 also holds.
Let now be a set system with properties 1, 2 and 3. Because of 1 and 3, holds: we have from which follows . Thus, from the union stability with respect to countable unions, we also get the union stability with respect to finite unions. Thus, is a ring. To show that is a -ring, let be an increasing set sequence in . Then, because of property 2, we have . That is, is closed with respect to taking limits of increasing set sequences and hence is a -ring.
Every finite ring (i.e. contains only finitely many sets) is a -ring: The second property in the definition is trivially satisfied, since there are only finitely many sets which can be joined, and is closed as a ring under finite unions.
If is a sequence of countable sets, then their union is also countable, so we have that .
If are countable, then because , their difference is also countable and so .
Example (Finite subsets)
By contrast, the set system of finite subsets of is indeed a ring, but not a -ring: For the sets we have that
Example (-ring of cuboids)
Consider the set system
of the axis-parallel cuboids in . (The intervals in the product may be open, half-open, or closed.) We already know the ring of cuboids in . It is defined as the ring generated by , that is, the smallest ring containing . In the same way one can consider the -ring generated by . Like the ring of cuboids, can be defined as the intersection of all -rings over containing the cuboids . (Compare also the article on generated -algebras).
While the ring generated by the cuboids contains only cuboid unions, the -ring additionally contains all sets which can be approximated by cuboid figures, such as the circle.
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