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[I posted this on MSE a while ago, but no answer was forthcoming.]

I am looking for a simple proof of the Lebesgue density theorem for $\Bbb{R}^n$. The Wikipedia page on the Lebesgue differentation theorem leads me to a proof of that more general theorem, but mentions that there are simpler proofs of the density theorem giving a reference to the book Measure and Category by Oxtoby. Because of the pandemic, I don't have access to a library just now and I can't find a simpler proof online except for the case $n = 1$. Can anyone give me other pointers to proofs of the Lebesgue density theorem for $\Bbb{R}^n$ that are simpler than the proof via the Lebesgue differentation theorem.

Note that https://math.stackexchange.com/questions/39090/where-to-find-a-proof-of-the-lebesgue-density-theorem asks about the general case, but the answer leads to a broken link to a paper by Faure. If that was Faure's paper in the American Mathematical Monthly from February 2002, then it only covers the case $n = 1$.

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    $\begingroup$ The Oxtoby reference covers only the $1$-dimensional case. $\endgroup$
    – Liviu Nicolaescu
    Commented Feb 10, 2021 at 23:17
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    $\begingroup$ I think you can find online Wheeden-Zygmund's Measure and Integral that has the simple proof of the differentiation theorem (in $\mathbb{R}^n$) via the Hardy-Littlewood maximal function and the Vitali covering lemma, and the corollaries. $\endgroup$
    – Pietro Majer
    Commented Feb 11, 2021 at 4:28
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    $\begingroup$ The proof in the first chapter of Elias M. Stein's "Singular Integrals and Differentiability Properties of Functions" is based on the Hardy-Littlewood maximal theorem and isn't exactly short, but it does get to the heart of the matter. It uses the Vitali covering lemma, as the one mentioned by Fedor Petrov above. (is it the same proof?) A particularly efficient exposition of it (in about two pages - including the proofs of the Hardy-Littlewood maximal theorem itself and of the Vitali covering lemma) can be found in the first chapter of C.D. Sogge's "Fourier Integrals in Classical Analysis". $\endgroup$
    – Pedro Lauridsen Ribeiro
    Commented Feb 11, 2021 at 5:30
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    $\begingroup$ (@PietroMajer, you beat me to it...) Another book that discusses it is Chapter 7 of Rudin's "Real and Complex Analysis". Of all three references I cited, I think Sogge does the best job, but it's all the same proof - I guess the one in Wheeden-Zygmund is also the same (is it the source of the others, perhaps?) $\endgroup$
    – Pedro Lauridsen Ribeiro
    Commented Feb 11, 2021 at 5:37
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    $\begingroup$ @RobArthan There seems to be an entire course here on the subject and similar matters terrytao.wordpress.com/2010/10/16/…. It includes step-by-step exercises that eventually lead to a proof. $\endgroup$
    – dohmatob
    Commented Feb 11, 2021 at 9:28

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