  This paper establishes connections between the group-Fourier transform and
the geometry of measures in the Heisenberg group. Firstly, it is shown that if
the Fourier transform of a compactly supported, finite, Radon measure is square
integrable, then the measure must have a square integrable density. If it's
Fourier transform is integrable, the the measure must have a continuous
density. In addition, an alternative formulation of the Fourier transform on
the Heisenberg group is used to show that energies of measures can be computed
via integrals on an appropriate frequency space. This in turns opens the
possibility of using Fourier methods in the computation of Hausdorff dimension
of sets.
