Correction to 'On the existence and uniqueness of certain measures’

M. Bruhat has kindly pointed out to me that in my paper (Hayman 1961) the arguments leading to lemma 6 and hence to theorems 3( a ) and 5 are not known to be valid for sets which are ‘ensembles analytiques-réels’, i.e. locally the zero set of a real-analytic function. A decomposition of such sets into a countable union of irreducible sets exists but the intersection of two distinct, irreducible sets , E 1 E 2 of dimension n — 1 may still have dimension n — 1, at least when E 1 , E 2 are non-compact, and if E 1 , E 2 are compact this question is an open problem. However, the results remain valid provided that ‘real analytic variety’ is reinterpreted everywhere as being an ‘ensemble réel C -analytique’ in the sense of Whitney & Bruhat (1959). An ‘ensemble réel C -analytique’ is the zero-set of a single function real-analytic in the whole space (Whitney & Bruhat 1959, prop. 10, p. 153). It is also the countable union of C -irreducible components (prop. 11, p. 155). Further, the intersection of two distinct C -irreducible sets of dimension n is a set of dimension less than n (corollary to prop. 12, p. 155). With these definitions and theorems the arguments leading to lemma 6 of my paper, and hence also theorems 3( a ) and 5 remain valid.

Sheets of Real Analytic Varieties

In a previous paper (4) the author worked out some results on the analytic connectivity properties of real algebraic varieties, that is to say, properties associated with the joining of points of the variety by analytic arcs lying on the variety. It is natural to ask whether these properties can be carried over to analytic varieties, since the proofs in the algebraic case depend mainly on local properties. But although this generalization can be carried out to a large extent, there are, nevertheless, difficulties in the analytic case, owing mainly to the fact (cf. 2, § 11) that a real analytic variety may not be definable by means of a set of global equations. Thus, although the general idea of the treatment given here is the same as in (4), some variation in the details of the method has proved to be necessary, and some of the final results are slightly weaker in form.