The existence of inductive limits in the category of (topological) measure spaces is proved. Next, permanence properties of inductive limits are investigated. If (X, , ) is the inductive limit of the measure spaces (X, , ), we prove, for 1 p 221E;, that LP(X, , ) is embeddible into the projectilimit of Lp(X, , ) in the category Ban, for p < , respectively in the category C* in the case p = +. As an application, we exten existence theorems of strong liftings to inductive limits.