THE SINGULARITY OF GAUSSIAN MEASURES IN FUNCTION SPACE.

In this note we give a tractable necessary and sufficient condition for mutual singularity which is concerned only with the given covariance functions. The method of proof depends on a generalization, first obtained by Kraft, of a theorem of Kakutani. We give a somewhat more general version of Kraft's theorem together with a simpler proof. Disciplines Statistics and Probability This journal article is available at ScholarlyCommons: http://repository.upenn.edu/statistics_papers/214 430 MATHEMATICS: L. A. SHEPP PiRoc. N. A. S. nation thus permitted is thought to be especially conducive to the process of directional selection and fixation of novel homozygous states (homoselection). Such states might serve as the basis for the "genetic revolutions" underlying species formation. 13 Summary.-New data are presented which indicate that two Nearctic species, Drosophila robusta and D. americana, are structurally homozygous on their extreme southern margins in central Florida. The same collecting sites are the apparent northern margin of the Neotropical species, D. acutilabella, which is similarly structurally homozygous. Like other species tending toward reduced polymorphism in Florida (D. euronotus, D. nigremelanica, and D. willistoni), the more central areas of these species show high polymorphism. Structurally monomorphic populations, especially if small, marginal and inbred, are considered ideal sites for species formation. We are grateful to Mrs. M. S. Carson and Mr. S. Prakash for aid in collecting specimens, and to Dr. H. D. Stalker for supplying wild-caught flies from his collections. Miss Mary Christine Steiner and Mrs. Jean S. Russell rendered valuable technical assistance. * Research supported by grants G20107 and GB-1607 of the National Science Foundation. 1 Carson, H. L., in Genetics and Twentieth Century Darwinism, Cold Spring Harbor Symposia on Quantitative Biology, vol. 24 (1959), p. 87. 2 Carson, H. L., Asilomar Symp., in press. 3 Carson, H. L., Advan. Genet., 9, 1 (1958). 4 Carson, H. L., in Population Genetics: The Nature and Causes of Genetic Variability in Populations, Cold Spring Harbor Symposia on Quantitative Biology, vol. 20 (1955), p. -276. Carson, H. L., and H. D. Stalker, Evolution, 1, 113-133 (1947). 6 Hsu, T. C., Texas Univ. Publ., 5204, 35 (1952). Carson, H. L., and W. C. Blight, Genetics, 37, 572 (1952). 8 Stalker, H. D., Genetics, 49, 669 (1964). 9 Ibid., 883 (1964). 10 Townsend, J. I., Jr., Evolution, 6, 428 (1952). 11 Heed, W. B., and J. S. Russell, Proc. Intern. Congr. Genet. 11th, 1, 139 (1963). 12 Heed, W. B., and N. B. Krishnamurthy, Texas Univ. Publ. 5914, 155 (1959). 13 Mayr, E., Evolution as a Process (London: Geo. Allen & Unwin, 1954), p. 157. THE SINGULARITY OF GAUSSIAN MEASURES IN FUNCTION SPACE