It is known that for basic linear fuzzy measures the Aumann and the Choquet integrals defined on a special class of fuzzy subsets of some Banach space commute. We characterize basic linear fuzzy measures by means of appropriate linear functionals, and consequently the relevant integral representation (by means of the Lebesgue integral) is introduced. As a corollary the well-known additivity of perimeters of convex subsets in the real plane is obtained.