We distinguish a class of unbounded operators in L r , r ≥ 1, related to the self-adjoint operators in L 2 . For these operators we prove a kind of individual ergodic theorem, replacing the classical Cesàro averages by Borel summability. The result is equivalent to a version of Gaposhkin's criterion for the a.e. convergence of operators. In the proof, the theory of martingales and interpolation in L r -spaces are applied. 2000 Mathematics Subject Classification: Primary 47A35, 60F15; Secondary 40G10, 47B40.