Finite-Horizon Structures III: Measure-Theoretic Realisation of the Projective Y-Structure

This article develops the measure-theoretic realization of projective Y-structure within the finite-horizon framework. Starting from a positive measurable representative of the homogeneous scalar quantity associated with a finite-horizon Y-object, the paper shows how this representative induces, relative to a chosen full-support Radon reference measure, an associated Radon measure whose density is given by the same structural scalar. The resulting construction is purely structural: the measurable representative is interpreted as a density of finite-horizon structural weight, not as a probability density. A central result is that the intrinsic measurable object determined by the framework is not, in general, a single distinguished measure, but a corresponding Y-measure class defined up to equivalence of reference measures. In this way, the measurable realization remains fully consistent with the projective character of the underlying Y-structure. The article further proves that once a reference measure is fixed, the associated measure is uniquely determined by the measurable representative. It also establishes that measurable Y-morphisms act on the associated measure representatives by the same constant rescaling factor that governs the transformation of the homogeneous scalar quantity itself. Together with the companion axiomatic and differential formulations, this work shows that the same projective finite-horizon organization admits categorical, differential, and measure-theoretic realizations. Measure and geometry therefore appear here as complementary manifestations of a single projective Y-structure. This article is self-contained at the measurable level while directly extending Finite-Horizon Structures I and II. Within the broader Ranesis program, it provides the measure-theoretic layer of the finite-horizon framework and completes the structural triad formed by its axiomatic, geometric, and measurable realizations.