Since the introduction of capacities by Choquet in \emph{Theory of Capacities} (1954), these set functions have played an important role in several areas of mathematics and its applications, including potential theory, decision theory, possibility theory, and models of uncertainty in economics and artificial intelligence. Their non-additive nature allows them to capture phenomena that cannot be described by classical measures.

In this direction, Cerdà, Martín, and Silvestre introduced capacitary function spaces, defining $L^p$-spaces of real measurable functions that are $p$-integrable with respect to a positive capacity. Later, Delgado and Sánchez Pérez introduced $L^1$-spaces associated with a vector capacity (2017).

Motivated by these developments, we introduce and study $L^p$-spaces associated with a vector capacity, which naturally extend the classical $L^p$-spaces defined with respect to vector measures. We analyze their basic structural properties and provide a general framework for the $p$-th power of a quasi-Banach function space over a $\sigma$-finite positive capacity space $(\Omega,\Sigma,\lambda)$. We then study the $p$-th powers of the $L^1$-spaces associated with a vector capacity, showing that this approach is equivalent to the previous one. This approach yields a unified setting for several constructions arising in the theory of integration with respect to non-additive set functions.

This work is carried out in collaboration with Celia Avalos Ramos and Enrique Alfonso Sánchez Pérez and forms part of the author’s doctoral research.