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Representation of maxitive measures: an overview
Idempotent integration is an analogue of Lebesgue integration where
-maxitive measures replace -additive measures. In addition to
reviewing and unifying several Radon--Nikodym like theorems proven in the
literature for the idempotent integral, we also prove new results of the same
kind.Comment: 40 page
Representation of non-semibounded quadratic forms and orthogonal additivity
In this article we give a representation theorem for non-semibounded
Hermitean quadratic forms in terms of a (non-semibounded) self-adjoint
operator. The main assumptions are closability of the Hermitean quadratic form,
the direct integral structure of the underlying Hilbert space and orthogonal
additivity. We apply this result to several examples, including the position
operator in quantum mechanics and quadratic forms invariant under a unitary
representation of a separable locally compact group. The case of invariance
under a compact group is also discussed in detail
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