Bands in $L_p$-spaces

For a general measure space $(Omega,mu)$, it is shown that for every band $M$ in $L_p(mu)$ there exists a decomposition $mu=mu+mu^{primeprime}$ such that $M=L_p(mu)={fin L_p(mu);f=0 mu^{primeprime}text{-a.e.}}$. The theory is illustrated by an example, with an application to absorption semigroups.