SOME REMARKS ON SUP-MEASURABILITY

Abstract

We give an example of a sup-measurable and non-measurable function in the general case associated with products of $\sigma$-ideals. We show that any behaviour of a function $F:{\mathbf{R}}^2\to \mathbf{R}$ with respect to $(L)$- and $(B)$-measurability, and $(L)$- and $(B)$-sup-measurability is possible. Continuum Hypothesis is replaced by weaker conditions. Finally, the notion of quasi-sup-measurability is considered.