The main result of this paper is a characterization of the boundedness of a fractional maximal operator from a weighted Lebesgue space
on a homogeneous type space
. The key ingredients of proof are earlier results of the authors and a generalization of the classical result of E. T. Sawyer
[Stud. Math. 75, 1-11 (1982; Zbl 0508.42023
)] to the context of homogeneous type spaces.