A fundamental description of continuum mixtures is presented. The mixture is decomposed into Borel sets associated with each constituent and the mixture is defined as the union of these sets. The decomposition is initiated in Euclidean one-space, motivated by the fact that the deformation of a continuum is ultimately described by the deformation of line elements. The decomposition is then extended to higher dimensions resulting in a volume fraction based theory for the kinematic description and the description of constituent and mixture properties. In particular, it is shown that the deformation gradient associated with the motion of any constituent can be decomposed into a constitutively prescribed, or deformation, component and another component referred to as the carrier component. The carrier arises because each constituent is embedded in others that also may deform and cause additional apparent deformation. The governing equations at both the constituent and mixture levels are developed using a general balance equation applicable to both cases. Some differences between the proposed theory and standard theories for continuum mixtures are discussed.