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Suppose $\mu$ is the Lebesgue measure defined on $\Bbb R^k$, I want to show that $\mu$ has some kind of linearity, which seems intuitively correct:

Suppose $A$ is a linear transformation on $\Bbb R^k$ (namely, a square matrix), then given any Lebesgue measurable subset $V$, $$\mu(AV)=|\det A|\mu(V),$$ where $AV:=\{Av\mid v\in V\}$.

I'm almost sure of it. But I couldn't find a proof about this property in any literature at hand.

I don't think the proof should be hard, though, if the elementary case has been proved. But that's exactly where I get stuck: how am I supposed to, for example, prove the $V=[0,1]^k$ case? Now if this can be proved, then I may just use elementary sets (unions of $k$-cells each two of which share an at most measure zero set) to approach $V$ and the result follows.

Could anyone help me on this part or give me other insights about the proof? Thanks in advance.

Caleb Stanford
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Vim
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  • If $A$ is invertible, then $\det A$ is the signed volume of the image of the unit cube by $A$ - this is also true if $A$ is not invertible. – Eman Yalpsid Apr 24 '16 at 12:39
  • @Andrew how to rigorously prove it, now that the "volume" is in the Lebesgue measure sense? – Vim Apr 24 '16 at 12:40
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    You're looking for a proof of the change of variables theorem for Lebesgue integrals. See, for example, this one – Ben Grossmann Apr 24 '16 at 13:13
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    This is proved in detail in Folland Real Analysis. Also in Rudin Real and Complex Analysis, I believe. – David C. Ullrich Apr 24 '16 at 13:43
  • @DavidC.Ullrich I have Rudin's R & C at hand but don't seem to find any related proof there. – Vim Apr 24 '16 at 14:08
  • Well, my Rudin is missing - maybe it's not there, maybe you haven't found it. It's in Folland. – David C. Ullrich Apr 24 '16 at 14:12
  • @Omnomnomnom thanks for the general case. – Vim Apr 24 '16 at 14:13
  • Your statement is generally a stepping stone for the general case – Ben Grossmann Apr 24 '16 at 14:21
  • It's in Rudin in the chapter on differentiation. – zhw. Apr 24 '16 at 16:08
  • It's not Chapter 7 of R&C. Chapter 2 discusses this in detail. It should be 2.20(e). – Ningxin Apr 24 '16 at 18:57
  • @QiyuWen 2.20 has only (a) to (d). – Vim Apr 25 '16 at 01:39
  • @Vim I just checked, and it appears that you are not reading the complete third version of his book. And you need to relate to 2.23 for the full result (or just prove it yourself, with the help of 2.20(e). It shouldn't be a challenge). – Ningxin Apr 25 '16 at 04:57
  • @QiyuWen I see. My roomie got a 3rd edition and I found it. It turns out that the result relies quite heavily on the Riesz representation theorem, though. – Vim Apr 26 '16 at 00:07
  • @6005 yeah I'm now sure it must've been asked before. – Vim Jul 20 '16 at 17:02
  • @Vim I believe it is an exact duplicate of that previous question I linked, with an answer. If you agree, can you confirm? There should be an option for you to immediately confirm this as a duplicate. +1 by the way, it's a good question. – Caleb Stanford Jul 20 '16 at 17:04
  • @6005 I think it's indeed a dupe. Thanks. I'll flag it as duplicate. – Vim Jul 20 '16 at 17:21

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