<p>The theory of classical Banach function spaces is associated with measures and measurable functions. We will extend the theory by replacing the measures with non-additive capacities, and the measurability of functions will also be removed. The function spaces of Lorentz type defined in terms of Choquet integrals associated with capacities will be addressed. The Köthe dual and block decomposition of such spaces are also discussed. As an application, we give a different viewpoint of the boundedness of Hardy-Littlewood maximal function on Choquet integral spaces.</p>