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Assume that $X$ is a Banach space of measurable functions for which Koml\'os'Theorem holds. We associate to any closed convex bounded subset $C$ of $X$ acoefficient $t(C)$ which attains its minimum value when $C$ is closed for thetopology of convergence in measure and we prove some fixed point results foraffine Lipschitzian mappings, depending on the value of $t(C)\in [1,2]$ and thevalue of the Lipschitz constants of the iterates. As a first consequence, forevery $L<2$, we deduce the existence of fixed points for affine uniformly$L$-Lipschitzian mappings defined on the closed unit ball of $L_1[0,1]$. Ourmain theorem also provides a wide collection of convex closed bounded sets in$L^1([0,1])$ and in some other spaces of functions, which satisfy the fixedpoint property for affine nonexpansive mappings. Furthermore, this property isstill preserved by equivalent renormings when the Banach-Mazur distance issmall enough. In particular, we prove that the failure of the fixed pointproperty for affine nonexpansive mappings in $L_1(\mu)$ can only occur in theextremal case $t(C)=2$. Examples are displayed proving that our fixed pointtheorem is optimal in terms of the Lipschitz constants and the coefficient$t(C)$.

Koml\'os' Theorem and the Fixed Point Property for affine mappings