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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).

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    Lebesgue theory still treats limits in Cauchy's sense, as arbitrarily close approximations rather than real or completed infinities. – Dan Hicks Feb 3 at 6:04
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    The "standard" approach to infinity in calculus and analysis is that expressed by Gauss : infinity is only a "facon de parler" that must be replaced with the formal rigorous notion of limit. – Mauro ALLEGRANZA Feb 3 at 9:35
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    Things are different in set theory, from Cantor and on, where infinite "objects" are treated mathematically and the concept of number is "stretched" to allows for infinite numbers. – Mauro ALLEGRANZA Feb 3 at 9:36
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    Lebesgue integral just uses a more flexible approximation process. When you are approximating your function by simple functions you have to choose 0 values for them at the tails, and successively move their edge points outward. This effectively packages the two limits Riemann takes into a single scheme (there is also a difference in approximating on finite intervals, but I assume that is beside the point). This is not salient to "completed infinity", which Cantor openly endorsed already, and the measure theory presupposes anyway. – Conifold Feb 3 at 11:04
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    Because Lebesgue first decomposes functions into positive and negative parts, and demands that both integrals exist. This is because he wants nice closure properties for integrable functions. The improper Riemann integrals that are not Lebesgue integrals are those where positive and negative parts are infinite. This is similar to conditionally convergent series, in the big picture they behave badly - rearranging the terms can produce any desired sum, as Riemann proved. Of course, one may well define improper Lebesgue integral, if one wants it. – Conifold Feb 4 at 3:56
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The Riemann integral doesn’t work well infinity as endpoints ...

In fact it does, you simply need to choose your space of functions carefully. Generally this is the space of continuous compactly supported functions. These are functions that are zero as you go towards infinity. The space of continuous rapidly decreasing functions is another.

It’s because the function is bounded in both directions that we can make sense of its area in a simple way, when it’s not, then other strategies need to be adopted. This is just as much a problem of the Lesbegue integral as it is for the Riemann integral.

The question of how to make sense of improper integrals has nothing to do with infinity - other than to notice that it is saying that the theory breaks down here - and it is all about how to understand the meaning of integration and this makes it a question of mathematics and how to understand the maths in mathematics rather than philosophy.

edit

The other small point to make, but relevant to the discussion, is that the Lesbegue integral is also defined in a two step process, essentially because it allows infinity as a value. But since infinity - infinity is undefined the standard method of defining the Lesbegue integral calculates it first for positive part of a function and then for the negative part and then subtracts the two.

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