Geodesic
Discrete measures with compact nowhere dense support, orthogonal to rational functions
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part VIII, Tome 73 (1977), pp. 7-15
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In this paper we prove the existence of nonzero measures of the sort mentioned in the title. In particular, we prove that any set whose closure contains interior points supports such a measure. 
@article{ZNSL_1977_73_a0,
     author = {A. B. Aleksandrov},
     title = {Discrete measures with compact nowhere dense support, orthogonal to rational functions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {7--15},
     year = {1977},
     volume = {73},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1977_73_a0/}
}
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A. B. Aleksandrov. Discrete measures with compact nowhere dense support, orthogonal to rational functions. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part VIII, Tome 73 (1977), pp. 7-15. http://geodesic.mathdoc.fr/item/ZNSL_1977_73_a0/