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First let $\mathcal{B},\mathcal{L}$ be the Borel and Lebesgue $\sigma$-algebras, respectively. I've been searching for my final project of real analysis on how to prove that $\mathcal{B}\varsubsetneq\mathcal{L}$ and I think i got the general idea of the prove:

You take the cantor function $\mathit{c}:[0,1]\to[0,1]$ and construct $f=\mathit{c}+I$ ($I$ is the identity function), then you analyze $f$ and conclude that $f$ is a homeomorphism.

After that, you observe that every lebesgue measurable set of non-zero Lebesgue measure contains a Lebesgue non-measurable subset and that this homomorphism $f$ "sends" Borel sets to Borel sets (most people mention a more general result of this and apply it to $f$).

We take $f(\mathcal{C})$, where $\mathcal{C}$ is the Cantor set, which is a Lorel set (then Lebesgue measurable, of measure $0$),and show that $f(\mathcal{C})$ is a Borel set, then Lebesgue measurable of measure 1, so we know there exist $K\subset f(\mathcal{C})$ not Lebesgue measurable.

But $f^{-1}(K)\subset \mathcal{C}$, so it's measurable of measure $0$ but not a Borel set, because otherwise it would make $K$ to be Borel too, reaching a contradiction.

My question is if there's a formal article I could use as a reference for my project, since I've only found blogs and informal articles where the people don't exactly justify all these steps.

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    $\begingroup$ When you write "homomorphism", what you really mean is "homEomorphism". $\endgroup$
    – PhoemueX
    Dec 3, 2019 at 17:26
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    $\begingroup$ The proof I knew was to describe Borel sets via transfinite recursion, observe that there are $\beth_1$, and then observe that all subsets of Cantor's set are Lebesgue measurable (and therefore there are $\beth_2$ Lebesgue measurable sets). $\endgroup$
    – user239203
    Dec 3, 2019 at 17:26
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    $\begingroup$ The google search "measurable" + "not Borel" + "Lebesgue" brings up a lot things worth looking at. For specific published references, since I tend to give lots of these in my answers/comments, try this search. $\endgroup$ Jan 4, 2020 at 9:29

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See for example the Appendix C of Frank Burk's Lebesgue Measure and Integration: An Introduction, available online at the Wiley Online library.

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