  Given a semiring with a preorder subject to certain conditions, the
asymptotic spectrum, as introduced by Strassen (J. reine angew. Math. 1988), is
a compact Hausdorff space together with a map from the semiring to the ring of
continuous functions, which contains all information required to asymptotically
compare large powers of the elements. Compactness of the asymptotic spectrum is
closely tied with a boundedness condition assumed in Strassen's work. In this
paper we present a generalization that relaxes this condition while still
allowing asymptotic comparison via continuous functions on a locally compact
Hausdorff space.
