A conjecture proposed by Naber says that if a sequence of $n$-dimensional non-collapsed manifolds $M_i$ Gromov-Hausdorff converges to $X$ with uniform lower curvature bound, then as a measure, the scalar curvature $scal_i dvol_{g_i}$ converges to a locally finite measure $\mu = R \, d\mathcal H^n +\Phi\, d\mathcal H^{n-1} + \Theta\, d\mathcal H^{n-2}$ on $X$. We will show that the integral of the scalar curvature on the smooth part of an Alexandrov space is locally finite and this affirms Naber's conjecture for the smooth part $R\, d\mathcal H^n$. We will also discuss the relation between this result and some conjectures by Yau and Gromov.