  We study band-dominated operators on (subspaces of) $L_p$-spaces over metric
measure spaces of bounded geometry satisfying an additional property. We single
out core assumptions to obtain, in an abstract setting, definitions of limit
operators, characterizations of compactness and Fredholmness using limit
operators; and thus also spectral consequences. In this way, we recover and
unify the classical and recent results on limit operator techniques, but also
gain new insights and are able to treat further applications.
