On the Egoroff property of pointwise convergent sequences of functions

Abstract:

The space $\mathcal {L}(x)$ of real-valued functions on $X$ has the Egoroff property if for any $\{ {f_{nk}}\}$ such that $0 \leqslant {f_{nk}}{ \uparrow _k}f$ (for every $n$), there exists ${g_m} \uparrow f$ such that, for each $m$ and $n$, ${g_m}{ \leqslant _{nk}}$ for some $k$. We show that $\mathcal {L}(X)$ has the Egoroff property if and only if the cardinality of $X$ is smaller than the minimum cardinality of an unbounded family of functions from the set of natural numbers to itself. Therefore, the statement that there is an uncountable set $X$ such that $\mathcal {L}(X)$ has the Egoroff property is independent of the axioms of set theory.