Fibers of $L^{\mathrm{\infty }}$ algebra

It is shown that Gelfand transforms of elements $f\in \text{lmu}$ are constant at almost every fiber ${\mathrm{\Pi }}^{-1}(\{x\})$ of the spectrum of $\text{lmu}$ in the following sense: for each $f\in \text{lmu}$ 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 ${\mathrm{\Pi }}^{-1}(\{x\})\cap U$.