Why is the Daniell integral not so popular?

The Riemann integral is the most common integral in use and is the first integral I was taught to use. After doing some more advanced analysis it becomes clear that the Riemann integral has some serious flaws.

The most natural way to fix all the drawbacks of the Riemann integral is to develop some measure theory and construct the Lebesgue integral.

Recently, someone pointed out to me that the Daniell integral is ‘equivalent’ to the Lebesgue integral. It uses a functional analytic approach instead of a measure theoretic one. However, most courses in advanced analysis do not cover the theory of the Daniell integral and most books prefer the Lebesgue integral.

But since these two constructions are equivalent, why do people prefer the Lebesgue integral over the Daniell integral?

Answer

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.

Attribution
Source : Link , Question Author : gifty , Answer Author : Michael Greinecker

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