We prove that if the Fourier transform of a compactly supported measure is in $L^2$ of a half-space, then the measure is absolutely continuous to Lebesgue measure. We then show how this result can be used to translate information about the dimensionality of a measure and the decay of its Fourier transform into geometric information about its support.

Pour toute courbe rectifiable du plan, nous d\'emontrons la formule de Cauchy relative \`a sa longueur. La formule est donn\'ee sous deux formes: comme int\'egrale de la variation totale des projections de la courbe dans les diverses directions et comme int\'egrale double du nombre de rencontres de la courbe avec une droite quelconque du plan. We give a general proof of the Cauchy formula about the length of a plane curve. The formula is given in two ways: as the integral of the variation of orthogonal projections of the curve, and as a double integral of the number of intersections of the curve with an arbitrary line of the plane.