Lebesgue measurability of separately continuous functions and separability

It is studied a connection between the separability and the countable chain condition of spaces with the $L$-property (a topological space $X$ has the $L$-property if for every topological space $Y$, separately continuous function $f:X\times Y\to\mathbb R$ and open set $I\subseteq \mathbb R$ the set $f^{-1}(I)$ is a $F_{\sigma}$-set). We show that every completely regular Baire space with the $L$-property and the countable chain condition is separable and construct a nonseparable completely regular space with the $L$-property and the countable chain condition. This gives a negative answer to a question of M.~Burke.