We propose a formalism (joint with José Iovino) of measure and integration of functions in the framework of real-valued logic. In contrast to existing such frameworks that either rely on nonstandard analysis, impose strict pointwise bounds on functions, or abstract away the underlying measure theory, our setup is meant to be user-friendly for mathematicians working in (standard) analysis or probability. In our integration structures, classical functions peacefully coexist with function-like objects that, from a classical perspective, have “escaped off” to become supported at infinity, or have concentrated as point masses. As a case study, we present a simple proof (both conceptually and technically) of the stability of Orlicz function spaces. We also discuss the suitability of our framework as an alternate foundation to construct “randomized structures” à la Keisler.