The restricted content and the d-dimensional Analyst's Travelling Salesman Theorem for general sets

In his 1990 paper, Jones characterized subsets of rectifiable curves in R2 via a multiscale sum β-numbers, which measure how far given set deviates from straight line at each scale and location. This characterization was extended by Okikiolu to Rn Schul Hilbert space. Recently, there has been some interest characterizing higher dimensional surfaces Rn. Using variant Jones' β-number introduced Azzam Schul, Villa gave lower regular certain class topologically stable – 2004 paper David these new β-numbers. this we remove the regularity condition prove an analogous result for general d-dimensional To do this, introduce restricted content, assigns ‘mass’ any subset (even sets with zero Hausdorff measure), use it define that is defined