Notation. Tasks, task laws, inference protocols, and utility [ftip-001N]

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Write \(\mathsf T\) for a task interface, \(\mathcal X_{\mathsf T}\) for its instance set, and \(\mathcal O_{\mathsf T}\) for its candidate-outcome set. A task instance is \(x\in \mathcal X_{\mathsf T}\) and a candidate outcome is \(o\in \mathcal O_{\mathsf T}\). The symbol \(\mu _{\mathsf T}\) denotes a probability law on instances. Equip the instance and outcome sets with declared sigma-algebras; \(\mu _{\mathsf T}\) is a probability measure on the instance sigma-algebra.

Fix an evaluation-resource dimension \(m_{\mathrm {eval}}\geq 1\) and write \(\mathcal B_{\mathrm {eval}}=\mathbb R_+^{m_{\mathrm {eval}}}\) with its coordinatewise order. An evaluation inference protocol is written \(\mathsf I\); its declared evaluation budget is \(b_{\mathrm {eval}}\in \mathcal B_{\mathrm {eval}}\). Its inference-seed set is \(\Omega _I\), with realized seed \(\xi \in \Omega _I\). An evaluator's random-seed set is \(\Omega _E\); its realized seed is \(\omega \in \Omega _E\). It returns utility \(u_{\mathsf T}(x,o;\omega )\in \mathbb R^d\). The dimension \(d\geq 1\) is declared by the evaluation: \(d=1\) gives scalar utility, while \(d>1\) retains several outcomes or costs without an implicit weighting. Equip both seed sets with declared sigma-algebras and \(\mathbb R^d\) with its Borel sigma-algebra. Products below carry the product sigma-algebra; finite or countable discrete spaces may use all subsets. A seed space alone does not specify its sampling law.