Let $a\in \mathbb{R}\setminus \mathbb{Q}$. I want to find a Lebesgue measurable function $f:[0,1)\to\mathbb{R}$ such that for all $m\in \mathbb{N}$, the functional equation $$mf(x)=h(x+a)-h(x) \pmod 1$$ does not have a measurable solution $h:[0,1)\to \mathbb{R}$.
Motivation: The main motivation is related to the following question Ergodicity of a skew product. This question gives a general criterion for ergodicity of a skew product considered. I want a specific example of such a skew product, which boils down to my question above.
Thanks in advance!