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This theorem characterizes those functions 4 E L. (oo, so) of the form +(x) = j (x) a.e. in terms of a continuity condition on the linear functional defined by F(f)= f f(x) b(x)dx for feL1(o,oo). Given a subset N c V(oo, cc) it is possible to ask whether additional continuity conditions on F can be found which, combined with the above condition, are necessary and sufficient in order that +(x) (x) a.e. for some tie N. The following theorem by A. C. Berry [1] illustrates such a condition when N is the class absolutely continuous functions of bounded variation. Here we define

Fourier transforms of certain classes of integrable functions