Mercer's Theorem for Vector-Valued Reproducing Kernel Hilbert Spaces in Kaplansky-Hilbert Modules over $L_{\infty}(\Omega)$

The study presents a vector-valued extension of the classical Mercer theorem within the framework of reproducing kernel Hilbert spaces defined over Kaplansky-Hilbert modules associated with the algebra of essentially bounded measurable functions. The analysis focuses on a partial integral operator with a positive definite kernel depending on a measurable parameter, and establishes the equivalence of three fundamental properties: the completeness of the system of eigenfunctions in the corresponding vector-valued space, the injectivity of the adjoint embedding operator, and the existence of a pointwise spectral decomposition of the kernel in terms of the eigenvalues and eigenfunctions of a parameterized family of operators. The proof relies on constructing an isometric isomorphism between the Kaplansky-Hilbert module and the space of measurable sections of a Hilbert bundle, thereby reducing the problem to the application of the classical Mercer theorem on each fiber of the bundle. Furthermore, a formalism of vector-valued lifting is developed to guarantee the coherence of inner product structures between the original module and its bundle representation.