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In this work we show that an $n$-dimensional Borel set in Euclidean $N$-spacewith finite integral Menger curvature is $n$-rectifiable, meaning that it canbe covered by countably many images of Lipschitz continuous functions up to anull set in the sense of Hausdorff measure. This generalises L\'{e}ger'srectifiability result for one-dimensional sets to arbitrary dimension andco-dimension. In addition, we characterise possible integrands and discussexamples known from the literature.  Intermediate results of independent interest include upper bounds ofdifferent versions of P. Jones's $\beta$-numbers in terms of integral Mengercurvature without assuming lower Ahlfors regularity, in contrast to the resultsof Lerman and Whitehouse.

Integral Menger Curvature and Rectifiability of $n$-dimensional Borel sets in Euclidean $N$-space