  In this paper we generalize Bochkariev's theorem, which states that for any
uniformly bounded orthonormal system $\Phi$, there exists a Lebesgue integrable
function such that the Fourier series of it with respect to system $\Phi$
diverge on the set of positive measure. We characterize the class of variable
exponent Lebesgue spaces $L^{p(\cdot)}[0;1]$, $1<p(x)<\infty$ a.e. on [0;1],
such that above mentioned Bochkarev's theorem is valid.
