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Article 48

Notes on the (PVB) and (PAC) functions

An. St. Univ. Ovidius Constanta 6 (1998), no. 2, 53-58.

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

 

Abstract

This paper deals with Sarkhel's conditions (PAC) and (PVB).

From Example 1 (that uses the Cantor ternary set and the Cantor ternary function), it follows that (PVB) is strictly contained in [VBG].

Then there are given given some characterizations of the (PAC) functions, using Lusin's condition (N) and Foran's condition (M):


Let F:E → R,  E is a closed set. The following assertions are equivalent:

  • F is(PAC) on E;
  • F is [VBG] and (N) on E;
  • F is (PVB) and (N) on E;
  • F is [(PVB)G] and (N) on E;
  • F is [(PAC)G] on E;
  • F is [VBG] and (M) on E;
  • F is (PVB) and (M) on E;
  • F is [(PVB)G] and (M) on E.