Bands in $L_p$-spaces

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

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