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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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