A generalized semigrand formalism for polydisperse fluids is presented and is used to derive a thermodynamic consistency equation. In the infinitely polydisperse limit—corresponding to a flat distribution of chemical potential differences—a characteristic parameter is eliminated, and the description of the mixture is greatly simplified. In the case of infinitely polydisperse hard spheres, the absence of a characteristic diameter implies that all quantities must scale to the density, which provides the only length. This leads to an exact equation of state which, remarkably, is PV/NkBT=4/3 at all densities. The treatment is generalized, to show that there exists a whole family of stationary composition distributions which have invariant compressibility factors. Monte Carlo simulation is used to verify these results, and applications to other potentials are discussed. Infinitely polydisperse fluids provide a convenient starting point for new mixture theories.