Analytic integration of metric-valued functions in Lipschitz free spaces

We develop an integration theory for functions taking values in a metric space. Following a Bochner-type construction, we define the concept of free integral as an element of the Lipschitz-free space $\mathcal{F}(M)$. We establish the main properties of this integral, including duality formulas, and the study of the resulting space of free integrable functions. We also cover when the metric space is a Banach space: in this setting, the free integral has an interpretable decomposition generalising the Bochner integral. We then connect the free integral with the geometry of $\mathcal{F}(M)$ by showing that it always produces convex integrals of molecules. This allows to study extremal properties within the unit ball of $\mathcal{F}(M)$. Finally, we provide a detailed example to illustrate the framework we develop.