  This paper extends the Lebesgue property and (weak) $G$-completeness to
generalized quasi-uniform spaces. It investigates the connections between
completeness, (weak) $G$-completeness, and the Lebesgue property of the product
of generalized quasi-uniform spaces with those of the component spaces. It has
been observed that the related behaviors differ for the Lebesgue property in
contrast to completeness and (weak) $G$-completeness.
