In Hohenberg-Kohn density functional theory, the energy E is expressed as a unique functional of the ground state density rho(r): E = E[rho] with the internal energy component F(HK)[rho] being universal. Knowledge of the functional F(HK)[rho] by itself, however, is insufficient to obtain the energy: the particle number N is primary. By emphasizing this primacy, the energy E is written as a nonuniversal functional of N and probability density p(r): E = E[N,p]. The set of functions p(r) satisfies the constraints of normalization to unity and non-negativity, exists for each N; N = 1, ..., infinity, and defines the probability density or p-space. A particle number N and probability density p(r) functional theory is constructed. Two examples for which the exact energy functionals E[N,p] are known are provided. The concept of A-representability is introduced, by which it is meant the set of functions Psi(p) that leads to probability densities p(r) obtained as the quantum-mechanical expectation of the probability density operator, and which satisfies the above constraints. We show that the set of functions p(r) of p-space is equivalent to the A-representable probability density set. We also show via the Harriman and Gilbert constructions that the A-representable and N-representable probability density p(r) sets are equivalent.