Systems of functions which are complete in measure

In this note it is proved that if a complete orthonormal system $\{\varphi_n\}$ in $L_2[0,1]$ contains a subsystem $\{\varphi_{n_k}\}$ of a lacunary order $p>2$, then for some bounded measurable function $h(x)$ the system $\{h(x)\varphi_n(x)\}_{n\ne n_k}$ is complete in $L_2[0,1]$.