  In this paper a weighted variable exponent Lebesgue spaces $L_{p(x), \omega}$
for $0< p(x)< 1$ is investigated. We show that this spaces is a quasi-Banach
spaces. Note that embedding theorem between weight variable Lebesgue spaces is
proved. In particular, we show that $L_{p(x), \omega}(\Omega)$ for $0< p(x)< 1$
isn't locally convex. Also, in this paper a some two-weight estimates for Hardy
operator are proved.
