Singularizing preserving countable additivity quantum channels on quantum measurable cardinals

We investigate the structural and dynamical properties of a class of quantum channels on von Neumann algebras induced by averaging over operator groups via the Pettis integral. Utilizing the classical Yosida--Hewitt decomposition, we focus on the interplay between the set-theoretic properties of the underlying space and the topological nature of the resulting quantum states. We establish a suffucient condition under which a channel preserves $\sigma$-additivity and exhibits a singularizing property, completely suppressing the normal component of any incoming state. In conjunction with the theory of Ulam real-valued measurable cardinals, this framework reveals a novel phenomenon: the existence of quantum channels that transform normal states into singular yet strictly $\sigma$-additive states. Furthermore, we analyze the structural constraints on the preservation of state purity imposed by the cardinality of the continuum, and extend our constructions to invariant measures on groups and their unitary representations, establishing the convergence of their Ces\`aro averages in the strong operator topology.