Definition. Evaluation utility [ftip-001T]
Definition. Evaluation utility [ftip-001T]
With the notation of Notation [ftip-001N], an evaluation utility is a declared measurable function \[ u_{\mathsf T}:\mathcal X_{\mathsf T}\times \mathcal O_{\mathsf T}\times \Omega _E\longrightarrow \mathbb R^d, \] where \(\Omega _E\) is the evaluator's measurable seed space. The evaluator applies it only to admissible pairs from Definition [ftip-001O]. A deterministic evaluator is the special case in which the value does not depend on \(\omega \in \Omega _E\).
Binary pass-fail evaluation takes \(d=1\) and values in \(\{0,1\}\). A cumulative return is another scalar case. Vector utility keeps qualities such as correctness, safety, latency, and tool cost distinct until a later rule declares how to compare them.
Whenever an expected utility vector is used, its composite evaluation random variable must be measurable and absolutely integrable in every coordinate under the declared joint law. Thus its expectation lies in \(\mathbb R^d\). Bounded measurable utility is a sufficient specialization; so is finite support for the complete evaluation random variable with finite values on that support. Merely taking a finite real value at each seed is insufficient for an unbounded evaluator. The scalar protocol comparison in Convention [ftip-005D] states its joint law explicitly.