Thanks again, but how to construct a bi-measurable function which takes null sets to null sets?

Thanks, but why all $L^\infty(\Omega)$ are isomorphic? if $\Omega_1$ and $\Omega_2$ are topologically isomorphic, the isomorphism between $L^\infty(\Omega_1)$ and $L^\infty(\Omega_2)$ is trivial. P.s. the criterion in Dunford and Schwartz's book seems very hard to apply, sad...