Ricci tensor on ${\mathrm{R}\mathrm{C}\mathrm{D}}^{\ast }(K,N)$ spaces

We obtain an improved Bochner inequality based on the curvature-dimension condition ${\mathrm{R}\mathrm{C}\mathrm{D}}^{\ast }(K,N)$ and propose a definition of $N$-dimensional Ricci tensor on metric measure spaces.