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.
