On the Extension of a Theorem of Caratheodory in the Theory of Fourier Series

The aim of this paper is to prove certain results which are converses of the following theorem in the theory of Fourier Series.

THEOREM A . Let {f_k(x)}, 0≤x≤2π, k-1, 2.........

be a uniformly bounded sequence of measurable functions convergent almost everywhere to a measurable function F(x).