Why is the Daniell integral not so popular?

Solution 1:

The following excerpt is from Measure Theory Vol 2 by Vladimir Bogachev:

In the middle of the 20th century there was a very widespread point of view in favor of presentation of the theory of integration following Daniell’s approach, and some authors even declared the traditional presentation to be “obsolete”. Apart the above-mentioned conveniences in the consideration of measures on locally compact spaces, an advantage of such an approach for pedagogical purposes seemed to be that it “leads to the goal much faster, avoiding auxiliary constructions and subtleties of measure theory”. In Wiener, Paley [1987, p. 145], one even finds the following statement: “In an ideal course on Lebesgue integration, all theorems would be developed from the point of view of the Daniell integral”. But fashions pass, and now it is perfectly clear that the way of presentation in which the integral precedes measure can be considered as no more than equivalent to the traditional one. This is caused by a number of reasons. First of all, we note that the economy of Daniell’s scheme can be seen only in considerations of the very elementary properties of the Lebesgue integral (this may be important if perhaps in the course of the theory of representations of groups one has to explain briefly the concept of the integral), but in any advanced presentation of the theory this initial economy turns out to be imaginary. Secondly, the consideration of measure theory (and not only the integral) is indispensable for most applications (in many of which measures are the principal object), so in Daniell’s approach sooner or later one has to prove the same theorems on measures, and they do not come as simple corollaries of the theory of the integral. It appears that even if there are problems whose investigation requires no measure theory, but involves the Lebesgue integral, then it is very likely that most of them can also be managed without the latter.

Solution 2:

Just a partial answer... As Qiaochu remarks, the "standard viewpoint" will not change without considerable impetus. The fact that some variant of "Lebesgue" integration is not "perfect" doesn't matter: it is "good enough".

Further, I would claim that, in fact, "the integral" people mostly use is not so much formally defined by any particular set-up, but is characterized, perhaps passively, in a naive-category-theory style (or, those might be my words) by what properties are expected. That is, for many purposes, we truly don't care about the "definition" of "integral", because we know what we expect of "integrals", and we know that people have proven that there are such things that work that way under mild hypotheses...

For all its virtues (in my opinion/taste), this "characterization" approach seems harder for beginners to understand, so the "usual" mathematical education leaves people with definitions...

My own preferred "integral" is what some people call a "weak" integral, or Gelfand-Pettis integral (to give credit where credit is due), characterized by $\lambda(\int_X f(x)\,dx)=\int_X \lambda(f(x))\,dx$ for $V$-valued $f$, for all $\lambda$ in $V^*$, for topological vector space $V$, for measure space $X$. This may seem to beg the question, but wait a moment: when $f$ is continuous, compactly-supported, and $V$ is quasi-complete, locally convex, widely-documented arguments (e.g., my functional analysis notes at my web site) prove existence and uniqueness, granted exactly existence and uniqueness of integrals of continuous, compactly-supported scalar-valued functions on $X$. Thus, whatever sort of integral we care to contrive for the latter will give a Gelfand-Pettis integral.

Well, we can use Lebesgue's construction, or we can cite Riesz' theorem, that every continuous functional on $C^o_c(X)$ (Edit: whose topology is upsetting to many: a colimit of Frechet spaces. But, srsly, it's not so difficult) is given by "an integral" (somewhat as Bourbaki takes as definition).

Either way, we know what we want, after all.

An example of a contrast is the "Bochner/strong" integral, which has the appeal that it emulates Riemann's construction, and, thus, directly engages with traditional ... worries? But, after the dust settles, there is still a bit of work to do to prove that the (as-yet-unspoken) desiderata are obtained.

Further, surprisingly often, in practice, the "weak" integral's characterization proves to be all that one really wants/needs! Who knew? :)

Solution 3:

People aren't going to switch to a new formalism unless they have a compelling reason to. Everyone already knows measure theory (including the professors teaching analysis courses). It is very powerful and suffices for many applications, so until someone convinces the mathematical community that an alternate theory of integration would fix their troubles, the mathematical community is not going to care. The perceived benefit of switching needs to outweigh the transition cost of learning a new formalism.

(I have heard of Mikusinski's approach to the integral, and in fact the first functional analysis course I took used this approach specifically to avoid measure theory. It was not enjoyable. The proofs of the basic theorems involve intricate manipulations of sequences of sequences and I don't remember any of them.)