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Article 33

Lusin's condition (N) and Foran's condition (M) are equivalent for Borel functions that are VBG on a Borel set

Real Analysis Exchange 23 (1997/8), no. 2, 477-490.

Mathematical reviews subject classification: 26A45, 26A46, 26A21, 26A30.

 

Abstract

Lusin's condition (N) has an important role in the theory of integration, since the classes of primitives for many nonabsolutely convergent integrals (Denjoy- Perron, Denjoy, α-Ridder, β-Ridder, Sarkhel-De-Kar) are contained in (N) ∩ VBG.
         
In Article 26 we showed that (N) ∩ VBG is a real linear space for Borel functions on Borel sets.

However Foran's condition (M), that strictly contains condition (N), seems to be more relevant to the theory of integral (see Real Functions - Current Topics).
         
In this paper we show that Lusin's condition (N) and Foran's condition (M) are equivalent for Borel functions that are VBG on a Borel set (see Theorem 2, (ii)).
         
In fact we prove stronger results (see Theorem 2, (i), (iii)), using conditions M and (N). These results are very useful to prove theorems of Hake-Alexandroff-Looman type (see for example Real Functions - Current Topics).
         

In the present paper we give some new characterizations of the conditions (M) and M.