  Kuelbs-Steadman spaces are introduced in this article on a separable metric
space with finite diameter and finite positive Borel measure. Kuelbs-Steadman
spaces of the Lipschitz type are also discussed. Various inclusion properties
are also discussed. In the sequel, we introduce HK-Sobolev spaces on metric
mesure 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.
