A refined version of the well-known Lebesgue decomposition theorem for positive measures is proven. The singular continuous part <$E mu sub roman sc> is shown to admit a unique decomposition into three mutually singular components, <$E mu sub roman sc~=~roman mu sub sc sup C~+~roman mu sub sc sup P~+~roman mu sub sc sup S>, of Cantor, Pratsiovytyi, and Salem types, respectively. This leads to a corresponding unique decomposition <$E sigma sub roman sc (A)~=~roman sigma sub sc sup C~union~roman sigma sub sc sup S~union~roman sigma sub sc sup P> of the singular continuous spectrum <$E sigma sub roman sc (A)> of a self-adjoint operator <$E A> on a Hilbert space <$E H>. Correspondingly, the singular continuous subspace <$E H sub roman sc (A)> of <$E A> admits the orthogonal decomposition <$E H sub roman sc (A)~=~H sub roman sc sup roman C~simbol e~H sub roman sc sup roman P~simbol e~H sub roman sc sup roman S> such that <$E A> is reduced by each of these subspaces, and the spectral measure <$E mu sub psi> associated with a vector <$E psi~symbol <174>~H sub roman sc sup roman C (A)> has only trivial components of P and S types, whereas the spectral measure <$E mu sub psi> associated with a vector <$E psi~symbol <174>~H sub roman sc sup roman P (A)> has only a trivial component of S-type.

• Кошманенко Володимир Дмитрович (1943–) (фізико-математичні науки)
• Торбін Григорій Мирославович (фізико-математичні науки)

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