Example of Non-Measurable Bijection from $\mathbb R \to \mathbb R$

Question

The title is fairly self explanatory. I am looking for an example of a function from $\mathbb R\to\mathbb R$ that is bijective, but not measurable with respect to Lebesgue measure, if such an example exists.

Answer 1

Take a none measurable subset $X\subset [0, 1]$ whose cardinal is the same as that of $[0, 1]$ - for example let $X$ be a non-measurable subset in $[0, \frac 12]$ union with $[\frac 12, 1]$. Let $f$ be a bijection from $X$ to $[0, 1]$. Now the cardinal of ${\mathbb R}-X$ equals to the cardinal of ${\mathbb R}-[0, 1]$ (both equals the cardinal of ${\mathbb R}$ since both contains an interval), so we can take $f: {\mathbb R}-X \to {\mathbb R}-[0, 1]$ to be a bijection. Fitting together we get the bijection $f:{\mathbb R}\to{\mathbb R}$ but $f^{-1}([0, 1])$ is not measurable.