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In an effort to find a principle underlying such generalizations, we were led to the theorem stated in the abstract which reduces certain questions of a.e. convergence for u.c. series in L to the same questions for orthogonal series.Recall that a series 27; in X is unconditionally convergent if the map T(otj) = 2 ajfj is continuous as an operator from lx to X. Further, since the natural injection of L into L, is continuous when/? > 1, an unconditionally convergent series in L with p > 1 may as well be considered to be an unconditionally convergent series in L,. We now are in a position to prove the theorem stated in the abstract.

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