Abstract. In this paper, we make a study of weighted locally convex spaces of measurable functions parallel to the studies of weighted locally convex spaces of continuous functions which has been a subject of intense research for decades. With Lp, 1 <= p < 1, spaces as our motivation, the completeness and inductive limits of those spaces are studied including their relationship with the weighted spaces of continuous functions leading to new results and generalizations of results true for Lp spaces.