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Let $X$ be a locally compact space. A lifting $\rho$ of $M_R^{\infty }(X,\mu )$ where $\mu$ is a positive measure on $X$, is almost strong if for each bounded, continuous function $f$, $\rho \left(f\right)$ and $f$ coincide locally almost everywhere. We prove here that the set of all measures $\mu$ on $X$ such that there exists an almost strong lifting of $M_R^{\infty }(X,|\mu \left|\right)$ is a band.