I know there exist (Lebesgue) measurable functions $f:\Bbb R\to\Bbb R$ such that $f\circ f$ is non-measurable (Here can be found an example).
Moreover, on can adapt this construction to find, for any $a\ne0$, a measurable function $f_a:\Bbb R\to\Bbb R$ such that $f_a(a\cdot f_a(x))$ is non-measurable.
My question is,
IS there a (Lebesgue) measurable function $f:\Bbb R\to\Bbb R$ such that for any $a\ne 0$ the function $f(af(x))$ is non-measurable?