Introduction. It is the purpose of this paper to develop a Lebesgue theory of integration of scalar functions with respect to a countably additive measure whose values lie in a Banach space. The class of integrable functions reduces to the ordinary space of Lebesgue integrable functions if the measure is scalar valued. Convergence theorems of the Vitali and Lebesgue type are valid in the general situation. The desirability of such a theory is indicated by recent developments in spectral theory. In §1 two criteria for the conditional weak compactness of subsets of the Banach space of countably additive measures on a c-field 2 are derived. Their force is sufficient to allow us to conclude, in §2, that if/z is a countably additive measure on 2 with values in a Banach space, then there exists a positive scalar measure ^ o n 2 with respect to which /x is ^-continuous (i.e., ^-absolutely continuous). This permits the development of the integration theory. As an example of an elementary application of the integration theory we give, in §3, the * Vector" generalization of the celebrated Riesz theorem on the representation of linear functional on a space of continuous functions. This yields representation theorems for the general, the weakly compact, and the compact operators on a space of continuous functions. Some of these results are related to theorems of Gelfand (6) and Grothendieck (7). Our notation and terminology are standard. We mention specifically that the weak topology of a Banach space 36 is the topology induced by its adjoint space 36* in the familiar fashion, even though 36 itself may be the adjoint of some other space. By the 36 topology of 36* we mean the topology on 36* which has as a typical neighbourhood of the origin, the set {x* 6 36* 1|#*(#*)I < 1» i = 1, . . . , n}.