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 $Rsim 0.3-0.6$, 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.
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