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I am applied science student who is self-studying the "Measure and Integral, An Introduction to Real Analysis, Second Edition" by Antoni Zygmund, and having a question about the Exercise 11 in Chapter 6.

Use Fubini's theorem to prove that $\int_{\mathbb{R^n}}e^{-|x|^2}dx = \pi^{n/2}$.

By polar coordinate change to evaluate the integral when $n = 1$, and applying Fubini's theorem to reduce the general $n$ into the case of $n=1$, this exercise is not hard. In fact I have evaluated this integral in my first-year undergraduate calculus course, under the framework of Riemann integral.

However, here is the question: so far it does not mention or prove the change of variables under Lebesgue's framework other than the case of nonsingular linear transformation in Exercise 20, Chapter 5. How do I justify the polar coordinate change? Or more generally, is there a theorem for change of variable for Lebesgue integral over a subset of $\mathbb{R^n}$?

I found that Exercise 11, Chapter 7, has a result concerning changes of variables in one-dimensional case, but it is not enough for polar coordinate change. Thus, I am curious why does the author not concern the change of variable, which in my experience is one of the most significant technique to evaluate an integral?

Thank you in advance.

Justin Lien
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  • Yes, there is a change of variables theorem. There's actually an "abstract" version valid for any measures, and then specializing to $\Bbb{R}^n$ one recovers the usual one (involving the Jacobian determinant). See this answer for a proof. You're right that change of variables is one of the most important results (but pretty tough to prove; my proof above invokes some measure-theory machinery, but conceptually simpler I think). Without polar coordinates, the only way I see is to use some complex analysis and relate this to the Gamma function. – peek-a-boo Aug 27 '21 at 07:56
  • I thought the easiest method is to write $|x|^2 = x_1^2 + \cdots + x_n^2$ and then use Fubini. Polar coordinates is not needed. – Arctic Char Aug 27 '21 at 08:14
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    @ArcticChar yes, but after Fubini, one has to calculate $\int_{-\infty}^{\infty}e^{-x^2},dx$, which is typically done by squaring this, and using polar coordinates in the plane to deduce it is $\sqrt{\pi}$ – peek-a-boo Aug 27 '21 at 08:16
  • @peek-a-boo I saw your proof in the link and I can understand the idea behind it even with my limited knowledge about abstract measure theory. Thank you very much for your help! – Justin Lien Aug 27 '21 at 09:51

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