A complete orthonormal system of divergence

A complete orthonormal system of functions {Θn}n=1∞,Θn∈L[0,1]∞, defined on the closed interval [0,1] is constructed such that ∑n=1∞anΘn diverges almost everywhere for any {an}n=1∞∉ l2. For the constructed system the following result is true: Corollary 1. Any nontrivial series in the system {Θn}n=1∞ which converges in measure to zero diverges almost everywhere.