This paper is concerned with a generalization of the concept of unitary equivalence of spectral measures on a Baer *-semigroup. A connection is made between abstract spectral measures, and three other distinct mathematical systems. Chapter II is devoted specifically to generalizing the concept of a spectral measure and to determining necessary and sufficient conditions for which two spectral measures will be unitarily equivalent. Chapter III discusses the problem of each (C(M) , qμ) being type I in terms of cycles, the basic elements of C(M). In Chapter IV it is shown that in a Loomis *-semigroup each type I (C(M) , qμ) will be type I homogeneous. Chapter V relates the study of unitary equivalence of spectral measures and the unitary equivalence of normal elements in a Finite Dimensional Baer *-algebra. / Ph. D.