# Formal article to read that the Borel $\sigma$-algebra is properly contained in the Lebesgue $\sigma$-algebra?

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.

• When you write "homomorphism", what you really mean is "homEomorphism". Dec 3, 2019 at 17:26
• 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).
– user239203
Dec 3, 2019 at 17:26
• 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. Jan 4, 2020 at 9:29