The Riemann integral itself doesn't work with infinity (±∞) as “endpoints”, you have to take a detour by calculating the integral for arbitrary endpoints ±z and then take the limit for z→∞, which makes it an “improper” integral.
In effect, with ℝ as domain of integration the following holds:
- With the improper Riemann integral you have to calculate two limits, a limit of a sequence and then the limit of a function towards infinity.
- But with the Lebesgue integral only one limit, that of a sequence, is necessary.
Still, not always does a Lebesgue integral exist when an improper Riemann integral exists, e.g. sin(x)/x with endpoints ±∞ (this is the ∞ - ∞ problem and can only occur for functions f where the integral of min(f, 0) is already -∞).
Now, in the context of the famous quote:
... first of all I must protest against the use of an infinite magnitude as a completed quantity, which is never allowed in mathematics. The Infinite is just a mannner of speaking, in which one is really talking in terms of limits, which certain ratios may approach as close as one wishes, while others may be allowed to increase without restriction. – C. F. Gᴀᴜss
can we say that the Lebesgue integral does achieve what Gᴀᴜss declares to be impossible?
Obviously, infinity is approached differently with the Lebesgue integral than with the improper Riemann integral.
But how? What is the exact philosophical difference?
PS: sorry, if that question gets closed here as off-topic because it's too mathematical, it will also certainly get closed as off-topic on math.SE for not being mathematical enough (the same happened with my “fair lottery of the natural numbers” question).