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Let      M B {M_B}       denote Lebesgue measure on the open unit ball,      B B      , in complex      N N      -space and let      M ( B ) M(B)       denote the space of Borel measures on      B B      . The volume Poisson kernel      χ  : B ¯  ×  B →  ( 0 , ∞  ) \chi :\overline B \times B \to (0,\infty )       is defined and then we prove Theorem.  If       μ  ∈  M ( B ) \mu \in M(B)        and if       μ  ♯  ( w ) = ∫  B χ  ( z , w ) d μ  ( z ) {\mu ^\sharp }(w) = {\smallint _B}\chi (z,w)d\mu (z)        is pluriharmonic in       B B      ,  then       μ  ♯  ∈  L ′ ( M B ) {\mu ^\sharp } \in L’({M_B})        and       μ  = μ  ♯  ⋅  M B \mu = {\mu ^\sharp } \cdot {M_B}      .

An F. and M. Riesz type theorem for the unit ball in complex $N$-space