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 PROJECTIONS AND SLICES OF MEASURES Commun. Korean Math. Soc.Published online November 16, 2020 Bilel Selmi and Nina Svetova Faculté des Sciences de Monastir, Petrozavodsk State University Abstract : We consider a generalization of the $L^q$-spectrum with respect to two Borel probability measures on $\mathbb{R}^n$ having the same compact support, and also study their behavior under orthogonal projections of measures onto $m$-dimensional subspace. In particular, we try to improve the main result of Bahroun and Bhouri \cite{1}. In addition, we are interested in studying the behavior of the generalized lower and upper $L^q$-spectrum with respect to two measures on sliced'' measures in an $(n-m)$-dimensional linear subspace. The results in this article establish relations with the $L^q$-spectrum with respect to two Borel probability measures and its projections and generalize some well-known results. Keywords : Orthogonal projection, Dimension Spectra, Sections, Convolutions MSC numbers : 28A12, 28A78, 28A80, 31E05 Full-Text :