# On the $ω$@-primitive

In: Mathematica Slovaca, vol. 51, no. 4
Zbigniew Duszyński - Zbigniew Grande - Stanislaw P. Ponomarev

## Details:

Year, pages: 2001, 157 - 162
In this paper we continue some results of [KOSTYRKO, P.: Some properties of oscillation, Math. Slovaca 30 (1980), 157–162]. It is shown that given a nonnegative, upper semicontinuous (USC) function $f:X\to \overline {\Bbb R}$ where $X$ is a massive'' metric space, there is a function $F:X\to {\Bbb R}$ (which we call an $ω$@-primitive for $f$) whose oscillation equals $f$ everywhere on $X$. Moreover, $F$ could always be found in at most Baire class two. In particular, the $ω$@-primitive could be written in a simple form whenever $f$ is finite. Namely, $F=fφ$, where $φ$ is the characteristic function of an $Fσ$@-set or that of a $Gδ$@-set. Except massiveness'', no other assumptions concerning metric spaces are made. Our main tool is Teichmüller-Tukey's lemma.
Duszyński, Z., Grande, Z., Ponomarev, S. 2001. On the $ω$@-primitive. In Mathematica Slovaca, vol. 51, no.4, pp. 157-162. 0139-9918.
Duszyński, Z., Grande, Z., Ponomarev, S. (2001). On the $ω$@-primitive. Mathematica Slovaca, 51(4), 157-162. 0139-9918.