Abstract: Suppose that Ω is a domain of Cⁿ, n≥1, E⊂Ω closed in Ω, the Hausdorff measure H^2n-1 (E) = 0 , and ƒis holomorphic in Ω\E . It is a classical result of Besicovitch that if n=1 and ƒ is bounded, then ƒ has a unique holomorphic extension to Ω. Using an important result of Federer, Shiffman extended Besicovitch’s result to the general case of arbitrary number of several complex variables, that is, for  n≥1 . Now we give a related result, replacing the boundedness condition of ƒ by certain integrability conditions of ƒ and of ∂²ƒ/∂Ζ²_j, j=1,2,K ,n. 

Keywords: Holomorphic function, subharmonic function, Hausdorff measure, exceptional sets.