  We consider both the minimisation of a class of nonlocal interaction energies
over non-negative measures with unit mass and a class of singular integral
equations of the first kind of Fredholm type. Our setting covers applications
to dislocation pile-ups, contact problems, fracture mechanics and random matrix
theory. Our main result shows that both the minimisation problems and the
related singular integral equations have the same unique solution, which
provides new regularity results on the minimiser of the energy and new
positivity results on the solutions to singular integral equations.
