# Lebesgue measurable set that is not a Borel measurable set

exact duplicate of Lebesgue measurable but not Borel measurable

BUT! can you please translate Miguel's answer and expand it with a formal proof? I'm totally stuck...

In short: Is there a Lebesgue measurable set that is not Borel measurable?

They are an order of magnitude apart so there should be plenty examples, but all I can find is "add a Lebesgue-zero measure set to a Borel measurable set such that it becomes non-Borel-measurable". But what kind of zero measure set fulfills such a property?

• See also this thread – t.b. May 4 '12 at 17:47
• @TheChaz because nobody would ever see that comment (except for people randomly looking for this...) – example May 4 '12 at 17:48
• It would notify Miguel and Jonas... – The Chaz 2.0 May 4 '12 at 17:50
• Prove: there are $\mathfrak{c} = \#\mathbb{R}$ Borel measurable sets. Every subset of the Cantor set is Lebesgue measurable. There are $2^\mathfrak{c} = \#P(\mathbb{R})$ subsets of the Cantor set. – t.b. May 4 '12 at 17:52
• Also relevant: math.stackexchange.com/questions/70880/… – user83827 May 4 '12 at 18:52

## 1 Answer

Let $\phi(x)$ be the Cantor function, which is non-decreasing continuous function on the unit interval $\mathcal{U}_{(0,1)}$. Define $\psi(x) = x + \phi(x)$, which is an increasing continuous function $\psi: [0,1] \to [0,2]$, and hence for every $y \in [0,2]$, there exists a unique $x \in [0,1]$, such that $y = \psi(x)$. Thus $\psi$ and $\psi^{-1}$ maps Borel sets into Borel sets.

Now choose a non Borel subset $S \subseteq \psi(C)$. Its preimage $\psi^{-1}(S)$ must be Lebesgue measurable, as a subset of Cantor set, but it is not Borel measurable, as a topological mapping of a non-Borel subset.

• This is. ${}{}{}{}$ – leo May 5 '12 at 0:36
• Why is it possible to choose a non Borel subset $S\subset \psi(C)$ ? $C$ is the Cantor set, right ? – Hua Mar 13 '17 at 10:15
• @bridger because $\psi(C)$ has positive measure and every set with positive Lebesgue measure contains a non-measurable set. Since Borel sets are measurable, the non-measurable set contained in $\psi(C)$ must be non-Borel. Now consider its preimage under $\psi$, you get a null set. So, it's Lebesgue measurable but it is not Borel because $\psi$ and $\psi^{-1}$ map Borel sets into Borel sets. – stressed out Feb 10 at 11:38