A note on almost strong liftings

Ionescu-Tulcea, C.; Maher, R.

Annales de l'Institut Fourier, Tome 21 (1971) no. 2, pp. 35-41.

Soit $X$ un espace localement compact. Un relèvement $\rho$ de $M_R^{\infty }(X,\mu )$, où $\mu$ est une mesure positive sur $X$, est presque fort si pour toute fonction continue et bornée $f$, $\rho \left(f\right)$ et $f$ coïncident localement $\mu$-presque partout. On démontre ici que l’ensemble des mesures $\mu$ sur $X$ telles qu’il existe un relèvement presque fort de $M_R^{\infty }(X,|\mu \left|\right)$, est une bande.

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.

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