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We give a new and very short proof of a theorem of Greiner asserting that apositive and contractive $C_0$-semigroup on an $L^p$-space is stronglyconvergent in case that it has a strictly positive fixed point and contains anintegral operator. Our proof is a streamlined version of a much more generalapproach to the asymptotic theory of positive semigroups developed recently bythe authors. Under the assumptions of Greiner's theorem, this approach becomesparticularly elegant and simple. We also give an outlook on severalgeneralisations of this result.

On a Convergence Theorem for Semigroups of Positive Integral Operators