Nonabsolute integrable function spaces on metric measure spaces

Kuelbs–Steadman spaces are introduced in this paper on a separable metric space having finite diameter together with a finite Borel measure. Kuelbs–Steadman spaces of Lipschitz type are also discussed. Various inclusion properties are also discussed. In the sequel, we introduce HK-Sobolev spaces on metric measure space which coincides with HK-Sobolev space in the Euclidean case. In application, we discuss the boundedness of Hardy–Littlewood maximal operator on Kuelbs–Steadman spaces and HK-Sobolev spaces over a metric measure space.