A measure differential inequality is established and is used to prove a result on the maximum solution, a comparison theorem and a uniqueness theorem of Perron type for abstract measure differential equations. Introduction. In previous papers [4], [5], the author introduced abstract measure differential equations and studied the problem of the existence of their solutions. As the differential and integral inequalities play a vital role in the theory of differential equations, it is essential to develop certain measure differential inequalities for the further study of (abstract) measure differential equations. This paper is an attempt towards this objective. In [4], [5], X is taken to be a real or complex linear space with Sx and S defined by (1.1) below. It may be noticed, however, that if X is normed x and Sx, Sx be defined by (1.2), all the results and proofs in [4], [5] still remain valid. In this paper Sx, S would be assumed to be defined by either of the equations (1.1) and (1.2) when X is normed. 1. Notation and preliminaries. Let X be a linear space over the field ?f where ?f is the set R of real numbers or the set C of complex numbers. For each x e X, we define S =ax: < a<1, S< = }ax: o < a < 1} iff= R; (1.1) sx t ax: 0 < lal < 1}, s x t ax: O < lal < 1} if =C. If X be normed, S and S may be assumed to have alternative meanings x x defined by (1.2) sx= f ex: llll < l1lxll X =IC lE X: jlell ' IlxliK Received by the editors September 26, 1973 and, in revised form, December 4, 1973. AMS (MOS) subject classifications (1970). Primary 34G05; Secondary 46G99.