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1* If S is a locally compact group and m e M(S) then the orbits Fm of m on all compact subsetsF of S [Fm: = {m*x\xeF)U{x*m\x eF), where x denotes the point mass at x] are weakly compact subsets of M(S) if and only if the restrictions SmlF of the orbit Sm of m on S to all compact subsets F of S [S m\F: ={j"Wj«eSm}] are weakly compact. The proof of this fact follows by observing that F'XK\\ = {xeS\Fx Π K Φ 0}] and KF"1 are compact as soon as both F and K are compact subsets of the group S. An arbitrary locally compact semigroup S may fail to have this compactness property and may [and actually does] give rise to two different subsets of M(S): namely, to L(S), the collection of all m e M(S) for which F\m\ is weakly compact (FQS, compact), and to Mβ(S), the collection of all m e M(S) for which S\m\\F is weakly compact (F £ S, compact). Elementaryproperties of L(S) can be found in e.g., [1], [2], [6], and [7] and of Me(S) in [4]. Although, in some respects Me(S} has better properties than L(S) [cf. [4], e.g., (5.2) and (5.3)] L(S) is a more obvious analogue of the group algebra than Me(S): L(S) is a two sided L-ideal

A note on measures on foundation semigroups with weakly compact orbits