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I am reading "Measure, Integration & Real Analysis" by Sheldon Axler.

I have read up to p. 59 of this book.  

I love this book and I guess this book is the best introductory book about Lebesgue integral in the world.

On the page which is nearly equal to $100\sqrt{2}$ (I guess it means p. 141), the author wrote the following:

To evaluate this integral, switch to the usual polar coordinates that you learned about in calculus ($d\lambda_2=rdrd\theta$), getting $\cdots$

So, I guess the author didn't write the change of variables theorem in this book.

Please tell me good introductory Lebesgue integral books which include a rigorous proof of the change of variables theorem.

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    $\begingroup$ Folland, Rudin. I’ve personally written an answer from a more high-tech perspective using the ‘abstract’ change of variables theorem (easy to prove), coupled with Lebesgue’s differentiation theorem and the Radon-Nikodym theorem in order to specialize to $\Bbb{R}^n$ (both of which are covered in both the books I suggested). Regarding “introductory Lebesgue integral books”, that’s kind of an oxymoron. Lebesgue integration is an advanced part of analysis, and the change of variables theorem is certainly advanced (to prove fully). $\endgroup$
    – peek-a-boo
    Apr 29, 2023 at 6:19
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    $\begingroup$ Also, Michael Taylor has a book, Measure Theory and Integration which is very nice, and where the theorem is proved along the lines I wrote above, and I liked it very much. $\endgroup$
    – peek-a-boo
    Apr 29, 2023 at 6:22
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    $\begingroup$ BTW, FYI, Royden has an improved fourth edition. $\endgroup$ Apr 29, 2023 at 6:39
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    $\begingroup$ I cannot say for that commenter but it got heavily revised by Fitzpatrick in the fourth edition: improved derivations, addition of contents, and more exercises. $\endgroup$ Apr 29, 2023 at 7:10
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    $\begingroup$ Priestley's Introduction to Integration proves the change of variables theorem for the Lebesgue integral and is an excellent book. $\endgroup$ Apr 30, 2023 at 18:13

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Here are three recommendations, each with its own chapter or section on change of variables and proofs of the stated theorems. Note, that each of them has its own specific setting of related theorems, its own kind of general treatment and preferences of applications. So, one or the other, or even all of them might be of interest.

An Introduction to Measure and Integration by Inder K Rana

Section 9.3 Change of variable formulas is devoted to the wanted topic, containing different theorems with proof. Here is

  • Theorem 9.3.9: Nonlinear change of variable formula: Let $T:V\to W$ be a bijective $C^1$-mapping from an open subset $V$ of $\mathbb{R}^n$ onto an open subset $W$ of $\mathbb{R}^n$ such that $J_T(x)\ne 0 \ \forall x\in V$. Then

    (i) if $E\in\mathcal{L}_{\mathbb{R}^n}$ and $E\subseteq V$, then $T(E)\in \mathcal{L}_{\mathbb{R}^n}$ and \begin{align*} \lambda(T(E))=\int_{E}\left|J_T(x)\right|\,d\lambda(x). \end{align*} (ii) if $f\in L_1(W)$, then $(f\circ T)\left|J_T\right|\in L_1(V)$ and \begin{align*} \int_{W}f(x)d\lambda(x)=\int_{V}\left(f\circ T\right)(x)\left|J_T(x)\right|d\lambda(x). \end{align*}

Measure Theory by D. L. Cohn. In section 6.1 Change of Variable in $\mathbb{R}^d$ the author deals with change of variable in $\mathbb{R}^d$ and with their relation to Lebesgue measure. The main theorem here is

  • Theorem 6.1.6: Let $U$ and $V$ be open subsets of $\mathbb{R}^d$, and let $T$ be a bijection of $U$ onto $V$ such that $T$ and $T^{-1}$ are both of class $C^1$. Then \begin{align*} \lambda(T(B))=\int_{B}\left|J_T(x)\right|\lambda(dx),\tag{4} \end{align*} and each Borel measurable function $f:V\to \mathbb{R}$ satisfies \begin{align*} \int_{V}fd\lambda =\int_{U}f(T(x))\left|J_T(x)\right|\lambda(dx),\tag{5} \end{align*} in the sense that if either of the integrals in (5) exists, then both exist and (5) holds.

A Concise Introduction to the Theory of Integration by D. W. Stroock provides the most general setting of the three books. Chapter V: Change of Variable is devoted to the theme.

  • Section 5.1: Lebesgue Integrals vs. Riemann Integrals relates integrals over an arbitrary measure space to integrals on the real line. The author states:

    • (not verbatim) ... The reason why it is often useful to make this change of variables is that the integral on the real line can often be evaluated as the limit of Riemann integrals to which all the fundamental facts of the calculus are applicable.
  • Section 5.2: Polar Coordinates a change of variables is examined for variables $x\in\mathbb{R}^n$ to the $(N-1)$-sphere $\{x\in\mathbb{R}^n:|x|=1\}$.

  • Section 5.3: Jacobi's Transformation and Surface Measure and Section 5.4: The Divergence Theorem treat change of variables in more general settings.

Hint: A thorough look at the table of contents of these books might be helpful in deciding which book might be most useful.

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  • $\begingroup$ epi163sqrt, Thank you very much for your detailed information. $\endgroup$
    – tchappy ha
    Apr 30, 2023 at 20:03
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    $\begingroup$ @tchappyha: You're welcome. $\endgroup$ Apr 30, 2023 at 20:06

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