A new method is suggested for obtaining Abel and Tauberian Theorems for integrals of the form ∞ 0 K t r dμ(t). It is based on properties of limit sets for measures. Accordingly, a version of Azarin's cluster set theory for Radon measures on the half-line (0, ∞) is created. Theorems of new sort are proved, in which the asymptotic behavior of the above integrals is described in terms of cluster sets for μ. With the use of these results and a stronger version (also proved in the paper) of Karleman's well-known analytic continuation lemma, the second Tauberian theorem by Wiener is refined considerably.2010 Mathematics Subject Classification. Primary 40E05; Secondary 30D20. Key words and phrases. Valiron's proximate order, Radon measure, Azarin's cluster set for a measure, Azarin's regular measure, Wiener Tauberian theorem.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 358 A. F. GRISHIN AND I. V. POEDINTSEVA Sometimes a more informative notation Ψ(K, r) will be used in place of Ψ(r).2. J(r) = 1 V (r) Ψ(r). 3. The measure s is defined by ds(t) = Ψ(t) dt. The symbol μ(t) will denote the distribution function of a measure μ, so that μ ((a, b]Licensed to Harvard Univ. Prepared on Sat Aug 1 16:29:38 EDT 2015 for download from IP 128.103.149.52. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ABEL AND TAUBERIAN THEOREMS FOR INTEGRALS 359The cluster set of a function f in the direction r → ∞ (i.e., the set of limits lim n→∞ f (r n ) as r n → ∞) will be denoted by L(f, ∞).A function f (r) is said to be compactly supportedThe properties of Fr[μ] have permitted us to prove several new theorems of Abel and Tauberian type for integrals of the form (1.2).Theorems of Abel type are those describing properties of Ψ if μ is given. Theorems of Tauberian type are those describing properties of μ on the basis of known properties of Ψ.Many well-known theorems of Abel type claim that Ψ(r) ∼ BV (r) whenever μ (t) ∼ A V (t) t (t → ∞). See, for instance, the books [2,5,6,7]. We state a simplest Abel type theorem of the present paper.Theorem 31. Let μ ∈ M ∞ (ρ(r)), and let K be a continuous compactly supported kernel on (0, ∞). ThenOther Abel type theorems obtained here are analogs of Theorem 31, which are proved under various restrictions on K and μ. Sometimes we lift the requirement that K be compactly supported. In other, more complicated cases, we also lift the requirement that K be continuous. In the general case, K is a Borel function on (0, ∞).An important distinction of the above results from the results known before should be mentioned. In the latter, the case of a regular measure μ was treated. In our results, a much wider class of measures is studied. In particular, this is M ∞ (ρ(r)) in Theorem 31.Without the assumption of continuity for K, the cluster set Fr[μ] with an arbitrary Radon measure μ does not determine the set L(J, ∞) any longer, as can be seen from Theorem 32.An important new result is the statement th...