A generalization of Lp-Brunn-Minkowski inequalities and Lp-Minkowski problems for measures

In this paper we extend the concepts of Lp-mixed volumes and Lp-surface area measures to Lp-mixed μ-measures and Lp-surface μ-area measures, respectively, for a measure μ on Rn. We give an integral representation and discuss some inequalities for the Lp-surface μ-area measure. The Lp-Minkowski and Lp-Brunn-Minkowski inequalities for measures are proved. Moreover, we also show the existence theorem of the related Lp-Minkowski problem for measures.