Use of granulometric size distributions to generate features for binary images was first studied by Matheron. Of late the method has been employed for numerous purposes including analysis of particle dispersion analysis of texture and image segmentation. Fundamental to the Matheron theory is his characterization of the most important class of granulometries the Euclidean granulometries. This characterization takes place in the context of his fundamental representation theorem for general binary tau-openings. The present paper provides a full extension of the Matheron theory of Euclidean granulometries to the gray scale.