Within the functional-integral formalism, for a half-filled band, we rigorously calculate the exchange-field probability distribution on a site as that which minimizes the grand potential including the classical fluctuation contribution. For large values of the intra-atomic Coulomb interaction $U$, a crossover temperature, instead of a transition temperature, roughly separates two regimes: a low-temperature regime exhibiting quasilocal moments and nonconducting states at the Fermi level, and a high-temperature regime characterized by spin fluctuations in a poor metal. A self-consistent expansion, up to the second order in external perturbations, allows us to define, on the same basis as the thermodynamics, the linear responses of the system. They all depend on the same integral kernel. Relations between the local exchange-field distribution and the local magnetization distribution are discussed. We estimate the magnetic susceptibility which evolves without any singularity from a Curie-Weiss-type behavior (large $U$, low temperature) to a renormalized Stoner-enhanced metallic behavior (small $U$, high temperature). The specific heat and the response to a Fermi-level change are also discussed. Finally, we extend the theory to account for antiferromagnetic ordering and for the occurrence of coupled amplitude and orientation fluctuations of the exchange field.