We provide new characterizations of Sobolev spaces that are true under some mild conditions. We study modified first order Sobolev spaces on metric measure spaces: \({{\,\mathrm{\textrm{TC}}\,}}\) -Newtonian space, \(\hat{{{\,\mathrm{\textrm{TC}}\,}}}\) -Newtonian space, and Ambrosio–Gigli–Savaré-like space. We prove that if the measure is Borel regular and \(\sigma \) -finite, then the modified \({{\,\mathrm{\textrm{TC}}\,}}\) -Newtonian space is equivalent to the Hajłasz–Sobolev space. Moreover, if additionally the measure is doubling then all modified spaces are equivalent to the Hajłasz–Sobolev space.
