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Article 1

On Foran's conditions A(N), B(N) and (M)

Real Analysis Exchange 9 (1984), 495-502

Mathematical reviews subject classification: 26A45, 26A46, 26A39

 

Abstract 

Two continuous functions F1 and F2 are constructed, satisfying Lusin's condition (N) and Foran's condition B(2) on C (C =  the Cantor ternary set), which satisfy Foran's condition A(N) on no portion  of C and for no natural number N.

Moreover F'1(x) = F'2(x) a.e. on [0,1], but F1 and F2 do not differ by a constant.

It is also shown that G(x) = F2(x) -½φ(x)  (φ = Cantor's ternary function), fulfils Foran's  condition (M), but does not fulfil Lusin's condition (N).

Such a function was already obtained by Foran in  An extension of the Denjoy integral  (Proc. Amer. Math. Soc., 49 (1975), 359-365), but in a more complicated way.