Abstract. Let $\Lambda $ be an open set in ${\msbm R},$ and define $$B^1(L^1_{\Lambda }({\msbm R})) =\{f(x) \in L^1({\msbm R}) | \hat f(t) = 0\hbox{ for all }t\notin\Lambda\}.$$ We define the set $\Lambda $ to be nicely placed if this class of functions is closed with respect to almost everywhere convergence. This term is analogous to that used by Godefroy and others in the setting of compact abelian groups. Our main result is that a set of the form $\Lambda = {\msbm R}^+\setminus K$ is nicely placed if $K$ is a closed set supporting a measure $\mu $ that has density on $K$ and whose Fourier-Stieltjes transform, $\hat\mu (t),$ belongs to the class $C_0({\msbm R}).$ In particular, if $K \subseteq{\msbm R}^+$ is any closed set of positive Lebesgue measure having density, then ${\msbm R}^+ \setminus K$ is nicely placed.

AMS Subject Classification (1991): 42A38

Received July 26, 1993. (Registered under 5554/2009.)