The inclusion of L sup p(mu) in L sup q(nu)

A theorem (the so-called main theorem) for the validity of the inclusion L sup p(mu) in L sup q(nu) is given, and proved by an application of the closed graph theorem. The main theorem is then applied to some interesting special cases, and concrete results are obtained including different versions of the results of Subramanian (1978) and Romero (1983).