Fibers of $L^{\infty}$ algebra

It is shown that Gelfand transforms of elements $f\in\lmu$ are constant at almost every fiber $\Pi^{-1}(\{x\})$ of the spectrum of $\lmu$ in the following sense: for each $f\in\lmu$ there is an open dense subset $U=U(f)$ of this spectrum having full measure and such that the Gelfand transform of $f$ is constant on the intersection $\Pi^{-1}(\{x\})\cap U$.