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*