Inventiones math. 51,239-260 (1979) Inve~l tio?les mathematicae 9 by Springer-Verlag 1979 On Invariant Measures, Minimal Sets and a Lemma of Margulis S.G. Dani * Tata Institute of Fundamental Research, Department of Mathematics, Homi Bhabha, Road, Bombay 400005, India Let G be a semisimple, or more generally a reductive, Lie group and let F be a lattice in G; i.e. G/F admits a finite G-invariant (Borel) measure. Let U be a horospherical subgroup of G; i.e. there exists g e G such that U = {x ~ G IgJx g- j ~ e asj ~ 0o } where e is the identity element in G. The action of U on G/F is called a horospherical flow. In [3] the author obtained a classification of all finite invariant measures of a certain class of horospherical flows. In the present paper we show that if F is an 'arithmetic' lattice then every locally finite ergodic invariant measure of the action of any unipotent subgroup (a horospherical subgroup as above is always unipotent) is necessarily finite. The first step is the following theorem. (0.1) Theorem. Let {ut}t~ ~ be a one-parameter group of unipotent matrices in SL(n, IR). 7hen every locally finite, ergodic, {u~}-invariant measure on SL(n, IR)/SL(n, 2g) is finite. Theorem 0.1 is closely related to the following result in [7] generally known as ' Margulis's lemma'. (0.2) Theorem. Let {ut}t~be as in Theorem 0.1. Then for any x~SL(n, IR)/SL(n, Z) the 'positive semi-orbit' {u,x[ t_->O} does not tend to infinity. That is, there exists a compact subset K of SL(n, 1R)/SL(n, 7l) such that {t >= O lUtXEK } is unbounded. Certainly, in view of Theorem 0.1 for any xeSL(n, IR)/SL(n,~.) the positive semi-orbit and the negative semi-orbit cannot both tend to infinity. For otherwise the 'time' measure along the orbit would be an ergodic, locally finite measure, which is invariant under the flow but not finite. On the other hand our proof of Theorem 0.1 involves finding a compact set K, for the given x, such that the set {t > Olu, x e K } has positive density (cf. Theorem 2.1). As we shall show in w3 in view of the individual ergodic theorem the last fact implies Theorem 0. l (cf. Theorem 3.2). Our proof of Theorem 2.t is modelled over M argulis's proof of Theorem 0.2. However, besides the stronger formulation, there is also a technical difference in our approach. We do not introduce any condition * Supported in part by a National Science Foundation grant (USA) 0020-9910/79/0051/0239/$04.40