Abstract: "This paper is grounded in the long-standing, purely abstract theory of integration with respect to nonlinear operator-valued measures μ :ℛ→N(S, F), which defines the integral operator Tf ≡ ∫︁ f dμ in full generality. By leveraging Pettis and Bochner Radon–Nikodým-type theorems and imposing natural assumptions on measurability and semivariation, we establish that this abstract integral can be represented by a classical scalar integral with a Carathéodory kernel φ : S×X →F. This representation, a core contribution of this work, translates the abstract operator integral into the more familiar form Tf = ∫︁X φ(f (y), y)dm(y). This approach not only provides a powerful analytical framework but also enables the direct use of standard tools for nonlinear integral operators. Moreover, by introducing a Lipschitz condition on the kernel, we derive the nonexpansive inequality ||Tf – Tg|| ≤ ||L||Lq ||f – g||Lp , thereby classifying T as a nonexpansive operator (or a strict contraction when ||L||Lq < 1). Consequently, this framework enables the direct application of classical fixed-point algorithms, such as Picard iterations for contractions and Krasnoselskii–Mann or Halpern schemes for nonexpansive mappings. This work bridges the gap between the abstract theory of operator-valued measures and the practical application of nonexpansive-type operator theory and algorithms."(...)