  We investigate some well-known problematic aspects of the single-jet
inclusive cross-section, specifically its non-unitarity and the possibly
related issue of apparent perturbative instability at low orders. We study and
clarify their origin by introducing possible alternative weighted definitions
of the observable which restore unitarity. We show that the perturbative
instability of the standard definition is an accidental artefact of the
smallness of the NLO $K$ factor which only manifests itself for values of the
jet radius in the range $R\sim 0.3-0.6$, and that its non-unitarity is
necessary in order to ensure cancellation of logs of the momentum cutoff used
in the jet definition. We also show that alternative unitary definitions do not
have better perturbative properties compared to the conventional non-unitary
definition, while suffering from lack of cancellation of large logs.
