ANALYTIC CAPACITY AND EQUICONTINUITY ANTHONY G. O'FARRELL In this paper we prove the following conjectures of Wang: Let 0(r) be a positive nondecreasing admissible function, let a be a point of a compact set X a C, let A n = {zeC : 2~" < \z-a\ < 2~ n + 1 } and suppose that where y denotes analytic capacity. Then (1) there exists a representing measure \i for a on K(X) such that \ (f)(\z-a\)~ i d\ji\(z) < oo, and (2) R(X) admits 0 as a modulus of approximate equicontinuity at a. Let X be a compact subset of the complex plane C. We denote by 0t{X) the space of all functions, continuous on C and analytic on a neighbourhood of X. We denote by R(X) the closure of M(X) with respect to the uniform norm on X: The uniform algebra R(X) has been studied intensively [1, 2, 3]. Apart from the natural questions of uniform rational approximation, interest has focused on R(X) as a source of counterexamples in the theory of Banach algebras. Indeed, the patho- logical behaviour of R(X) is so extraordinary that any positive result seems impressive. The most striking early result is Browder's metric density theorem [1, p. 177] which states that at any non-peak point aeX of R(X) the unit ball of R(X) is approximately equicontinuous, i.e., for each e > 0 the set {zeX : |/(z)-/(fl)| < e for all/e J?(X) with ||/|| x < 1} has full area density at a. This result was strengthened in various ways by the author [5, 6, 7], Wang [9, 10], 0ksendal [8] and Hayashi [4]. In particular, Wang [10] showed that at almost all nonpeak points, and for all a less than 1, all the functions belonging to the unit ball of R(X) satisfy a single Holder condition of order a on a set of full density. In his paper [10], Wang formulated three conditions, each of which might be interpreted as saying that the functions belonging to R(X) are of a certain degree of smoothness at the point a. To describe his conditions, we need some notation and terminology. An admissible function is a positive nondecreasing function <f>(r) on the interval (0, oo) such that the associated function t^(r) = /*/</> (r) is also nondecreasing, with ij/(0 + ) = 0. We denote the (inner) analytic capacity of an open set */" by [2, p. 196], and we set A n (a) = {zeC:2-"< \z-a\ <2~ n+1 } Received 12 November, 1977 [BULL. LONDON MATH. SOC, 10 (1978), 276-279]