A developmental process, independent of morphogenetic (shape) change, that is required for the cerebellar molecular layer to attain its fully functional state. The molecular layer is the outermost layer of the cerebellar cortex. It contains the parallel fibers of the granule cells, interneurons such as stellate and basket cells, and the dendrites of the underlying Purkinje cells. [ http://openisbn.com/search.php?q=0838580343&isbn=1/ http://purl.obolibrary.org/obo/go/references/0000021 GOC:dgh GOC:cls GOC:dph GOC:jid ]