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We present natural (invariant) definite and indefinite scalar products on the N = 1 superspace which turns out to carry an inherent Hilbert-Krein structure. We are motivated by supersymmetry in physics but prefer a general mathematical framework.

A supersymmetric integral theorem that extends results of Parisi, Sourlas, Efetov, Wegner, and others is rigorously proved. In particular, arbitrary generators are allowed in the integrand (instead of canonical ones) and the invariance condition is very much relaxed. The connection with Cauchy's integral formula is made transparent. In passing, the unitary Lie supergroup is studied by using elementary methods. Applications in the theory of disordered systems are discussed.qJi(X,S) =ai(x) +h i (x,s),degh i >2 and defineg(qJl, ... ,qJm) by a formal Taylor series expansion:

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A class of test functions ``minimal with respect to causality'' is introduced. The corresponding fields are called local. Tempered and strictly localizable fields are local, but there is a large class of fields that are local but not strictly localizable. For local fields, the analytic properties of vacuum expectation values are studied. The local fields that are not strictly localizable are characterized by an arbitrary fast increase of Wightman functions near the light cone. With an adequate definition of local commutativity, other properties of tempered and strictly localizable fields seem to remain valid.