It is known that each non-decreasing continuous function $\phi$ induces a $\sigma$-additive measure $d\phi$ such that $\int_0^1 f(x) d\phi(x)$ exists for every bounded real-valued Baire function $f$. This follows because every Borel set is measurable with respect to $d\phi(x)$, and for every Baire subset $A \subseteq [0,1]$ the characteristic function $\chi_A$ is a Baire function.
Conversely, suppose now that $A \subseteq [0,1]$ is such that $A$ is $d\phi$-measurable for every $\phi$ as above. Does it follow that $A$ is a Borel set?
(Question formulated by Prof. Jan-Erik Björk at Stockholm University)