Stieltjes Integration Meets Deep Learning: A Measure-Theoretic Framework for Actuarial Reserve Valuation

\begin{abstract}This research develops a novel measure-theoretic framework integrating Stieltjes integration with Bayesian deep learning for life insurance reserve valuation. Extending de Finetti's foundational work on subjective probability and Rudin's functional analysis, we reformulate actuarial reserves via Radon-Nikodym derivatives to jointly model financial and demographic risks. The framework combines measure-adapted neural operators, Lévy-process foundations, and quantum-inspired computational accelerations. Rigorous benchmarking on EIOPA mortality data shows that our Bayesian neural network reduces mean absolute error by 63\% compared to traditional methods, with a stochastic MLP variant achieving 75\% faster convergence. Theoretical contributions include a generalized Stieltjes operator for Lévy-type processes and information-geometric regularization of neural networks. Empirical results demonstrate 77\% KL divergence reduction and alignment with Lévy $\alpha=1.7$ tail behavior, confirming robustness to extremal events. A projected $O(\sqrt{N})$ complexity reduction via Grover's algorithm enables real-time reserving for large portfolios, offering transformative potential for Solvency II compliance.\end{abstract} Author affiliation: Independent Researcher; REPRISE – Register of Expert Peer Reviewers for Italian Scientific Evaluation, Ministry of University and Research (MUR), Italy Revised Version with Enhanced Educational Content and Clarifications “This is an updated version of DOI: 10.5281/zenodo.15477145”