The following remark is made by Halmos in his book [2, p. 29]. "Can an automorphism of a locally compact but noncompact group be an ergodic measure preserving transformation? Nothing is known about this subject. Only in the compact case has anything ever been done." The aim of this paper is to give an answer to this question of Halmos. We prove the following in this paper: (i) A continuous automorphism T of a locally compact group G which is not bicontinuous cannot be ergodic. (ii) A continuous automorphism T of a locally compact group G with a left invariant Haar measure pi (pi need not be right invariant) and which is not measure preserving cannot be ergodic. (iii) A continuous automorphism of a locally compact, noncompact abelian group is not ergodic. NOTATIONS. We follow [2 ] and [3 ] for notions in ergodic theory and measure theory. We follow [9] and [4] for notions in topological groups. All topological spaces are assumed to be Hausdorff in this paper. Unless otherwise stated the groups occurring in this paper are not assumed to be abelian. DEFINITION 1. Let G be a locally compact group. A continuous automorphism T of G is a one-to-one map from G onto itself which is continuous and is also an algebraic isomorphism. An autohomeomorphism or a bicontinuous automorphism T of G is an algebraic isomorphism from G onto itself such that both T and T-1 are continuous. REMARK. Some examples of locally compact groups G and continuous automorphisms of G which are not autohomeomorphisms are given by Robertson [8] in his thesis. To our best knowledge this is the first place where such an example is written. We give below an example of a continuous automorphism of a locally compact group which is not bicontinuous, based on an idea of K. H. Hoffman. This example shows the need of Lemma 1 below. EXAMPLE OF A CONTINUOUS AUTOMORPHISM OF A LOCALLY COMPACT GROUP WHICH IS NOT BICONTINUOUS. Let G1 be the circle group with the usual topology. Let F1 be the circle group with discrete topology. Let F1 = F2= ... = n = ... and let G1 = G2=G3= ... =Gn =