User:UKe-CH

I am a mathematician and I used to work in IT (information technology) living in Zurich (Switzerland), bilingue German-French with good English knowledge. My German website, which I mentioned here in the past, is now closed.

My activities in the Wikipedia(s) are mainly in German & French - clicking on one of these two language links shows my user-page in that language.

Wikipedia, NPOV and mathematics

One might think that there is no place for POV (Points Of View treated in an unilateral way) in mathematics. But although this may be essentially true for individual articles, it may be a failure when considering the collection of all (pure) mathematical subjects treated in the wikipedia site for some language. Mathematicians can have preferences for certain math. subjects and/or for different ways a subject may be treated, BTW often related to differences relative to languages. But such preferences should not induce an overall POV in Wikipedia's articles which I mention above as a failure. When differences in use between languages happen with definitions of math. expressions, one can consider this as a kind of false friends as occur between languages in ordinary vocabulary; and one has to accept such situations, there being practically no chance to end them, and these problems may be viewed in math. too in the usual way. Examples in math.: 1) the numbers defined as billion, trillion a.s.o. are not the same in (American and I guess usually British too) English and (at least most) continental European languages - for sure German and French. This example is at the same time one for general language, so it will have to be considered accordingly. 2) compact space VS. espace compact (F) - the word compact also has a special meaning in math. in comparison with general language including other sciences (!) but there is also a difference between mathematicians concerning their math. meaning (in F compact usually includes Hausdorff, while the same word in English is usually defined as is in F quasi-compact) - this kind of situation is often revealed in math. literature including wikipedias by saying "authors differ in their definitions of ..." or the like. But it is a fact that the dominating definition is clearly language dependent. This is not a major problem as long as readers have a good chance to discover the intended use when a such expression occurs in a wikipedia article about math's. What I am aiming at here is different and has to do with extensive treatment of math. theories. And I express my hope that this type of POV will also be eliminated in some not too far future (case by case) ...

In the following I want to discuss one - later may-be more - such a case of POV-like situation in math. as treated in the Wikipedias.

Two measure theories in competition

This is an example where what I say above about POV in math. subjects as treated in Wikipedia applies.

Mathematical measure theory (MT) exists in two versions which I will call for simplicity the classical one and the linear one.

The classical MT starts with a mapping from certain subsets of a given set X to one of the following sets: 1) that of the positive elements of the extended real number line i.e. the interval [0,+infinity] ... 0 is included! 2) the full ext. real line = [-infinity,+infinity] 3) the ordinary set of real numbers 4) the set of complex numbers. The choice 1) is normally discussed first and the others are often omitted completely, measure being then used as meaning positive measure i.e. one of those for that choice. Choice 2) has the consequence that an individual measure cannot have positive and negative infinite values. I am not sure that choice 3) is ever used. Choices 3) and 4) cannot have infinite values of a measure; they imply that a measure has to be bounded if the whole X is among the sets for which the measure is defined - this being the usual case. The collection of subsets of X used has normally to satisfy certain conditions, most frequently to be "closed" (i.e. stable) for the following operations: a) finite & countable set union, b) complementation with respect to X (sending a subset A to X\A). The mapping itself has to satisfy certain conditions, mainly countable additivity for disjoint sets. Once the measure of sets is established, integration (which can then be defined) leads to functions having as argument a function from X (or some part of X) to one of the sets mentioned above in choices 1) to 4) or, more generally, to real or complex vector spaces ... these functions are then normally linear.

The linear MT starts with a linear mapping sending functions defined in X and with complex values to complex numbers. Usually X has to be a locally compact Hausdorff topological space, the functions X -> C used as arguments for the lin. map are the continuous ones for which there is a compact subset K of X such that the function has value 0 at points not belonging to K (so-called functions with compact support). And the mapping itself has to be continuous with respect to a vector space topology defined as an inductive limit; this complicated feature can be replaced by a simple property of the lin. map that happens to be equivalent to said continuity. To get nontrivial examples of such mappings for X = R or some interval of R, one uses Riemann-like integrals, but one needs only to define these for regulated functions where it can be done in an especially easy way described in the article just mentioned & linked. The theory of Haar measure(s) on locally compact groups is often constructed directly in the frame of linear MT - and this includes the usual Lebesgue measure on R. Having established the basics of measures defined as such linear mappings, one then aims at extending the mapping to more general functions - dropping both continuity and compact support assumptions; this is similar to what is done in classical MT: one first considers positive measures for which - instead of so-called outer measures considered in classical MT one introduces superior integrals (my literal translation from a French expression) which makes it possible to define null sets, integrable functions and their integral (these steps are quite different from what is done in classical MT) for complex measures in the sense of linear MT. The measure of a subset of X is then defined as the integral of its characteristic function when this is integrable. In general, one has to distinguish between measurable and integrable sets / functions, the concept of measurability being introduced in the next step of the theory; for positive measures a measurable set / function may have infinite measure / integral, but this is usually undefined for nonpositive real or general complex measures. What I am describing here is the content of Bourbaki's Integration textbook up to chapter 4, somewhat simplified.

Now the classical MT and the linear MT both have advantages and disadvantages, not only in their methodological aspects, but also as their scope is concerned.

The first works well for positive measures, but has serious drawbacks for signed real or complex measures - to avoid these one has to restrict to bounded measures or to accept that the whole X may have no measure defined. There is no need for a topology on X, so when none is present, one talks about abstract measures. When in fact a topology on X is considered, an abstract measure may not have good topological properties. Extra axioms are needed - even for positive measures - to obtain better compatibility with the topology and those for a so called Radon measure result in measures that can be obtained via the linear MT (if I understand well). But the classical MT simplifies the treatment of images of (positive) measures. Abstract measures and non-Radon measures on a locally compact space extend the scope of MT beyond what delivers the linear MT ... but the question is, to which extent is this useful?

The linear MT has the advantage of its ... linearity, i.e. linear subspaces and linear mappings are much more central to it. The extension of the integral is a completion process (as for metric spaces) - it leads directly to the L^p spaces associated with a measure because these are completions of the space of continuous functions with compact support for vector space topologies - one per value of p in [1,+infinity] - this is the so called topology of convergence in the p-th mean. For p=1 the extension of the linear mapping from the continuous functions with compact support to the integrable functions is simply by continuity for this topology. The treatment of nonpositive real or complex measures works well for unbounded measures, so here the scope of MT extends beyond what you get with the classical MT ...

So the two versions of MT seem to be needed. The complexity of both implies that the links between them are perhaps still insufficient. Wikipedia as well as the majority of literature - may-be even in French - is currently too concentrated on the classical TM.