Periodica Mathematica Hungariea Vot 10 (1), (1979), pp. 9--I3 STABILITY OF LEBESGUE SPACES by B. P. DUGGAL (Nairobi) 1. Introduction Let G be a locally compact topological group with left Haar measure m. A Radon measure # on G is a positive measure defined on the Borel subsets of G such that # is locally finite and # is inner regular, i.e., for each Borel subset E of G, #(E) = sup (#(K) : K c E, K compact). If ] is a function defined on G, define the left translate, t~ f, of f by amount s E G to be the function (tJ) (y) -~/(s-~y); define the right translate, rs, analo- gously. Let Lp(,a), 1 ~ p ~ c~, denote the space of/t-equivalence classes of extended real-valued #-measurable functions / on G for which Ill p is #-integrable, and suppose that for each s E G, the translation operator ts (or rs) is continuous on Lp(It ), # a Radon measure, to itseff. Then what can we say about/~ ~. In this note was characzerize such measures #. It is shown that under suitable hypoth- eses on G, #, ts (or rs) the answer to the problem is that # is an m absolutely continuous left (or right) moderate measure. 2. Notation G will denote a locally compact (Hausdorff) topological group with left tt,~ar measure m. We denote by M(G) the space of all (positive) Radon meas- ures on G, and by Mb(G) the space of all bounded Radon measures on G. Le- # E M(G). We denote by ~p(/~) -~ ~p(G; it), 1 <~ p ~ c~, the space of #-measur- able (extended) real-valued functions f on G for which I/Is is #-integrable. Lp(/~), 1 < p ~ o% will denote the space of #-equivalence classes, modulo g-null sets, of functions /E s The left translate tsf (right translate rJ) of a function / by amount s E G will be defined as in the introduction. A.MS (MOS) subject classi#catione (1970). :Primary 46G10; Secondary 28A20, 28A70. Key words avid phrases. Locally compact group, left translation, loft quasi-invar-