# Learning the theory of the n-dimensional Riemann integral

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## Main Question or Discussion Point

I would like to learn (self-study) the theory behind the n-dimensional Riemann integral (multiple Riemann integrals, not Lebesgue integral). I am from Croatia and found lecture notes which Croatian students use but they are not suitable for self-study. The notes seem to be based on the book: J. E. Marsden, M. J. Hoffman, Elementary Classical Analysis but the book is expensive. Could you recommend me some books/notes/video lectures/... (languages: Croatian, English, German) ?

I also got some student notes and while going through them, two examples caught my eye. I want to learn how to make valid mathematical arguments for such examples.

Example 1.
Definition.
We say that C has an area if the function $\chi _C$ is integrable on C, i.e. on some rectangle that contains C. In that case, the area of C is $\nu (C) = \int _C \chi _C$ where:
$$\chi _C (x,y) = \begin{cases} 1, (x,y) \in C \\ 0, (x,y) \notin C \end{cases}$$and C is a bounded subset of $\mathbb{R}^2$.

The notes I've got ask such questions as "Does a disk have an area?", "Does a triangle have an area?". (in terms of the definition above)

Example 2.
$$C =\{ (x,x) | x\in\mathbb{R} \}$$
Claim: C has a (Lebesgue) measure of zero.
The notes say that the argument "C is just a rotated x-axis" is not valid because $d(k , k+1) = (k+1) - k = 1 < d(f(x_{k_1}), f(x_k))$ so we have a rotation and "stretching".

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mathwonk
Homework Helper
well i am puzzled that saying C is a rotated x axis is not valid. it certainly is a rotated x axis, but the map taking x to (x,x) does not happen to be the rotation map. still there is a rotation map from the axis onto this set.

second, there are free books available which explain the n dimensional "Riemann" integral, although just what that means may be variable since Riemann himself did not treat several variable integration, at least not in his famous paper on trigonometric series, where he defined his (one variable) integral.

The book Advanced Calculus by Loomis and Sternberg, although too abstract in many ways for convenient use, may be suitable for its theory of n dimensional "content" and integration (Riemann integration, not Lebesgue), and was free on Sternberg's website at Harvard, last time I looked.

The difference between Riemann and Lebesgue integration, or between content and measure, is the use of either finitely many or infinitely many rectangles to cover sets. For instance if you try to cover the rational points of the unit interval with a finite number of rectangles, their total length must be ≥ 1, but if you use an infinite number you can make the total length as short as you wish. So those rationals have content 1 but measure zero.

yes here it is: look at chapter 8:

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I would highly recommend measure theory if you’re up for it - the Lebesgue Theory is very nice and has many better properties than the Riemann integral.

However, since you did say you wanted to learn Riemann integration on Euclidean space, I’d recommend Duistermaat and Kolk’s Multidimensional Analysis, Volume 2.

WWGD
Gold Member
2019 Award
I think most Advanced Calc books contain this material.

Math_QED
Homework Helper
2019 Award
I think most Advanced Calc books contain this material.
Yeah, most treat it very badly. E.g. Rudin.

WWGD