Weak compactness and representation in variable exponent Lebesgue spaces on infinite measure spaces

Relative weakly compact sets and weak convergence in variable exponent Lebesgue spaces L-p(.) (Omega) for infinite measure spaces (Omega, mu) are characterized. Criteria recently obtained in [14] for finite measures are here extended to the infinite measure case. In particular, it is showed that the inclusions between variable exponent Lebesgue spaces for infinite measures are never L-weakly compact. A lattice isometric representation of L-p(.) (Omega) as a variable exponent space L-q(.) (0, 1) is given.