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