Characterizations of countably $n$-rectifiable radon measures by higher-dimensional Menger curvatures

We provide a characterization of countably n-rectifiable measures in terms $\sigma$-finiteness the integral Menger curvature. also prove that finiteness condition on pointwise curvature can characterize rectifiability Radon measures. Motivated by partial converse Meurer's work Kolasiński we under suitable density assumptions there is comparability between pointwise-Menger and sum over scales centered $\beta$-numbers at point.