This example provides a continuous function which is not Lebesgue measurable.
Where is my mistake in the following proof that such function does not exist ?
(1) B is the usual topology on $\mathbb{R}$ (generated by the euclidian metric)
(2) $\mathcal{B} = \sigma(B)$ is the Borel tribe (smallest tribe containing open sets in B)
(3) $\mathcal{N} = \{ A \in \mathcal{P}(\mathbb{R}), \exists N \in \mathcal{B} / \lambda(N)=0, A \subseteq N \}$ are the Borel negligible for the Lebesgue measure $\lambda$.
(4) $\mathcal{L} = \sigma(\mathcal{B},\mathcal{N})$ is the Lebesgue tribe, completion of $\mathcal{B}$
(5) $L= \{O - N, O \in B, N \in \mathcal{N} \}$ is the essential (non metric) topology
(6) We actually have that $\mathcal{L} = \sigma(L)$, $\mathcal{L}$ is the borelian tribe of $L$. See here and here, p291
(7) Proposition (vi) in the previous link (on the next page) states that the continuous functions are the same for the 2 topologies: $\mathcal{C}^0(B) = \mathcal{C}^0(L)$
(8) If $f$ is $L$-$L$ continuous, it is $\mathcal{L}$-$\mathcal{L}$ measurable.
So all together the initial example is continuous, hence $L$-$L$ continuous, hence $\mathcal{L}$-$\mathcal{L}$ measurable, hence measurable (that is $\mathcal{L}$-$\mathcal{B}$ measurable), which is in contradiction.
Did I make an obvious error somewhere ?
I am suspecting that (8) is false. Is it not true that $f$ $A$-$B$ continous $\Rightarrow$ $f$ $\sigma(A)$-$\sigma(B)$ measurable (on the tribes generated by the corresponding topology) ?
It is obviously true for the Borel version, so here I am assuming that if the preimage property holds on open sets of the topology (or any generating system of the tribe), it holds on the full tribe.
What would make it false in the case of Lebesgues ? Does it depend on the ordinal involved when generating the tribe from the open sets ? If so, what would be the Lebesgues hierarchy, compared to Borel_hierarchy ? Or is it the poor topological properties of $L$ that are problematic ?
And a bonus question: if (8) is false, are there any other notion of continuity (absolute-continuity maybe) that would ensure being $\mathcal{L}$-$\mathcal{L}$ measurable ?
Thank you !
