VITALI’S THEOREM WITHOUT UNIFORM BOUNDEDNESS

Let {fm}m≥1 be a sequence of holomorphic functions defined on a bounded domain D ⊂ Cn or a sequence of rational functions (1 ≤ deg rm ≤ m) defined on Cn. We are interested in finding sufficient conditions to ensure the convergence of {fm}m≥1 on a large set provided the convergence holds pointwise on a not too small set. This type of result is inspired from a theorem of Vitali which gives a positive answer for uniformly bounded sequence. 2010 Mathematics Subject Classification: Primary: 41A05, 41A63, 46A32.


Introduction
Let D be a domain in C n , {f m } m≥1 be a sequence of holomorphic functions defined on D. A classical theorem of Vitali asserts that if {f m } m≥1 is uniformly bounded on compact subsets of D and if the sequence is pointwise convergent to a function f on a subset X of D which is not contained in any complex hypersurface of D, then {f m } m≥1 converges uniformly on compact subsets of D. A striking feature of this theorem is that we may extend, in some sense, the function f which is a priori measurable and defined on a very small set, to a holomorphic function entirely on D. We note, however, that the assumption on uniform boundedness of {f m } m≥1 is essential.Indeed, using the classical Runge approximation theorem, it is possible to construct a sequence of polynomials on C that converges pointwise to 0 everywhere except at the origin where the limit is 1!In particular, the convergence is not uniform on any open set containing the origin.For details, see Example 1 in [6].We are concerned with finding analogues of the mentioned above theorem of Vitali in which the uniform boundedness of the sequence under consideration is omitted.A possible approach is to impose stronger mode of convergence and/or the size of X. Motivated by the problem of finding local conditions for single-valuedness of holomorphic continuation, Gonchar proved in Theorem 2 of [7] the following remarkable result.
Theorem 1.1.Let {r m } m≥1 be a sequence of rational functions in C n (deg r m ≤ m) that converges rapidly in measure on an open set X to a holomorphic function f defined on a bounded domain D (X ⊂ D) i.e., for every ε > 0 Here λ 2n is the Lebesgue measure in C n ∼ = R 2n .Then {r m } m≥1 must converge rapidly in measure to f on the whole domain D.
Much later, by using techniques of pluripotential theory, Bloom was able to prove an analogous result in which rapidly convergence in measure is replaced by rapidly convergence in capacity and the set X is only required to be compact and non-pluripolar (see Theorem 2.1 in [3]).More precisely, we have Theorem 1.2.Let f be a holomorphic function defined on a bounded domain D ⊂ C n .Let {r m } m≥ be a sequence of rational functions (deg r m ≤ m) converging rapidly in capacity to f on a non-pluripolar Borel subset X of D i.e., for every ε > 0 lim m→∞ cap({z ∈ X : |r m (z) − f (z)| 1/m > ε}, D) = 0.
Then {r m } m≥1 converges to f rapidly in capacity on D i.e., for every Borel subset E of D and for every ε > 0 Here cap(., D) denotes the relative capacity; a brief discussion of this kind of capacity as well as convergence in capacity will be given in the next section.Using a standard result which relates convergence in capacity and pointwise convergence (cf.Lemma 2.2), it is not hard to check that Theorem 1.1 follows from Theorem 1.2 (see Theorem 2.2 in [3]).The above theorems of Gonchar and Bloom are the main inspiration for our research.The first result of this paper, Theorem 3.1, states roughly that if a sequence of bounded holomorphic functions is convergence fast enough on a non-pluripolar set then the convergence is also fast uniformly on compact sets.Here the speed of approximation is measured in terms of the growth of the sup-norms of f m .The next result, much in the same spirit of Theorems 1.1 and 1.2, deals with a version of Vitali's theorem for a sequence of rational functions.In Theorem 3.4 we consider a sequence {r m } m≥1 of rational functions that is rapidly pointwise convergent on a Borel non-pluripolar subset of C n to a bounded measurable function.Under an additional condition that the degree of the denominator of r m tends to ∞ much less than m, we are able to show that the sequence {r m } m≥1 converges rapidly in measure entirely on C n to a measurable function F .The main result of the paper (Theorem 3.6), to some extent, is a generalization of Theorem 1.1 and Theorem 1.2 when the sequence {r m } m≥1 is supposed to converge rapidly to radial boundary values of some bounded holomorphic function f defined on a bounded domain D ⊂ C n .More precisely, we show that if the subset where the convergence occurs is not too small then the same type of convergence must hold inside the domain.Moreover, we also consider the convergence of {r m } m≥1 to f on affine subspaces of C n .As an illustration of this theorem, we establish in Proposition 4.2 an example of a bounded holomorphic function f on the unit disk ∆ and a sequence of rational functions {r m } m≥1 with poles lying outside ∆ such that {r m } m≥1 converges rapidly pointwise to f * , the radial boundary values of f , on a compact subset F ⊂ ∂∆ of positive length.Nevertheless, the function f does not extend holomorphically through any point of F .
Acknowledgements.This work has been started during a visit of the first named author at the Vietnam Institute for Advanced Mathematics (VIASM) in the winter of 2012.The paper was finally completed during a stay of the first and the second named authors at VIASM in the winter of 2014.We wish to thank VIASM for financial support and the warm hospitality.It is also our pleasure to thank Professor Pascal Thomas for a useful suggestion in the construction of Proposition 4.2.Last but not least, we are grateful to an anonymous referee for her/his constructive comments, especially for asking a clever question that leads to the current state of our main result (Theorem 3.6).The first named author is supported by the grant 101.02-2013.11from the NAFOSTED program.The second named author is also sponsored by the NAFOSTED program under the grant number 101.02-2014.01.

Preliminaries
For the reader convenience, we collect in this preparatory section necessary elements of pluripotential theory that will be needed later on.Let D be a domain in C n .An upper semicontinuous function u : D → [−∞, ∞) is said to be plurisubharmonic if the restriction of u on D ∩ l is subharmonic for every complex line l.The cone of plurisubharmonic function on D is denoted by PSH(D).A subset E of C n is said to be pluripolar if for every z 0 ∈ E there exists an open connected neighborhood U of z 0 and u ∈ PSH(U ), u ≡ −∞ such that u ≡ −∞ on E ∩ U .According to a classical theorem of Josefson, if E is pluripolar then there exists a plurisubharmonic function u which defined globally on C n such that u ≡ −∞ on E. Clearly a proper complex subvariety of D is pluripolar.On the other hand, it is not hard to show that any subset of C n with positive Lebesgue measure is not pluripolar.
In order to measure how close a Borel set to be pluripolar, following Bedford and Taylor (see [9, p. 120]) we let cap(E, D) be the relative capacity of a Borel subset E in D which is defined as It is well known that relative capacity enjoys some important properties such as sub-additivity and monotone under increasing sequences.Moreover, a deep result in Bedford-Taylor's theory states that pluripolar subsets of D are exactly subsets with vanishing relative capacity.We also frequently appeal to Bernstein-Walsh's inequality (see [11]) which states that if K, L are compact sets in C n and K is non-pluripolar, then there exists C K,L > 0 depending only on K and L such that for any polynomial p m in C n of degree at most m, We recall the following types of convergence of measurable functions which are essentially well known.
Definition 2.1.Let {f m } m≥1 , f be complex valued, measurable functions defined on a bounded domain D ⊂ C n .We say that the sequence {f m } m≥1 (i) converges in capacity to f on X if for every ε > 0 we have where X m,ε := {x ∈ X : |f m (x) − f (x)| > ε}; (ii) converges in capacity to f on D if property (i) holds true for every compact subset X of D.
We have the following relation between convergence in capacity and pointwise convergence.Lemma 2.2.Let {f m } m≥1 and f be complex valued, measurable functions defined on a domain D ⊂ C n .If {f m } m≥1 converges in capacity to f on a Borel subset X of D, then there exists a subsequence {f mj } j≥1 and a pluripolar subset E ⊂ X such that {f mj } j≥1 converges pointwise to f on X \ E.
Proof: Suppose that {f m } m≥1 is a sequence that converges in capacity to f on X.For every ε > 0, the hypothesis gives Hence, we can find a strictly increasing sequence {m k } such that cap(X m, 1  2 k , D) < 2 k and E := ∞ j=1 E j .By the sub-additive property of the relative capacity we obtain It follows that cap(E, D) = 0 and hence E is a pluripolar set.Now, for z ∈ X \ E, it is easy to check using the definition of E j that lim k→∞ f m k (z) = f (z).We are done.
We should mention that there exists a pointwise convergence sequence that contains no subsequence that converges in capacity.Indeed, let {A m } m≥1 be a sequence of pairwise disjoint subsets of the unit disk ∆ ⊂ C such that inf m≥1 cap(A m , ∆) > 0. Then the sequence of characteristic functions {χ Am } m≥1 provides the desired example.It should be remarked that we do not know if there exists a version of Egorov's theorem for convergence in capacity, i.e. every pointwise convergence sequence contains a subsequence that converges in capacity on some non-pluripolar subset.
We will frequently appeal to the following fundamental result of Bedford and Taylor (Theorem 4.7.6 in [9]) which partly explains the role of pluripolar sets in pluripotential theory.Then the set {z ∈ D : u(z) < u * (z)} is pluripolar.

Rapid convergence of holomorphic functions and rational functions
We start with the following generalization of Vitali's theorem mentioned in the introduction.Theorem 3.1.Let D be a domain in C n and {f m } m≥1 be a sequence of bounded holomorphic functions on D. Suppose that there exists an increasing sequence {α m } m≥1 of positive numbers satisfying the following properties: (i) (iii) There exists a non-pluripolar Borel subset X of D and a bounded measurable function f : X → C such that Then the following assertions hold: It follows from assumption (i) that {u m } m≥1 is a sequence of plurisubharmonic functions on D which is uniformly bounded from above on D.
We claim that lim For this, fix x ∈ X and ε ∈ (0, 1).By (3.1), there exists The claim follows by letting ε ↓ 0. Now we set Then u ∈ PSH(D).Furthermore, by the claim proven above and Lemma 2.3 we infer that u = −∞ on a non-pluripolar subset of X.Thus u ≡ −∞ on D. In particular u m converges pointwise to −∞ on D.
Next, by Hartogs' lemma we conclude that u m tends uniformly to −∞ on compact sets of D. It follows that, given a compact subset K of D, for every constant A > 0, there exists m(A) ≥ 1 such that Using (ii) and the triangle inequality we obtain Therefore, we may apply Cauchy's criterion to deduce that {f m } m≥1 converges uniformly on compact sets of D to a holomorphic function f .Moreover, by letting n → ∞ in the above inequality and observing that A can be made arbitrarily large, we get the desired rapid uniform convergence on K.The proof is complete.
Assume that there exists a non-pluripolar Borel subset X of C n and a measurable function f : X → C such that Then the following assertions hold: (a) {p m } m≥1 converges uniformly on compact sets of C n to a holomorphic function f .(b) For every compact subset K of C n we have lim m→∞ p m −f 1/m K = 0. Proof: Let D be a relatively compact domain in C n .According to Theorem 3.1, it suffices to prove that on the domain D, the sequence {p m } m≥1 satisfies the conditions given in Theorem 3.1.For this, we set By (3.2), we have N ≥1 X N = X.Since X is non-pluripolar, we infer that there must exist N 0 such that X N0 is non-pluripolar.Since X N0 is Borel, it must contain a non-pluripolar compact subset X .Since f is bounded on X, we obtain Thus, using Bernstein-Markov's inequality we get By simple estimates we get where C > 0 is a constant independent of m.Thus the sequence α m := Cm satisfies conditions given in Theorem 3.1.We are done.
The situation is technically complicated for sequences of rational functions because of the presence of poles sets.First, we need the following concept.
Definition 3.3.Let V be an algebraic hypersurface in C n and U be an open subset of C n .We define the degree of V ∩ U to be least integer d so that there exists a polynomial p of degree Now we are able to formulate our next result.Theorem 3.4.Let {r m } m≥1 be a sequence of rational functions on C n satisfying the following properties: (i) There exist a Borel non-pluripolar subset X of C n and a bounded measurable function f : (ii) For every z 0 ∈ C n , there exist an open ball B(z 0 , r), m 0 ≥ 1, and λ ∈ (0, 1) such that where V m denotes the pole sets of r m .
Then there exists a measurable function F : For the proof we need the following elementary fact.
Proof: By the assumption we have By Weierstrass's test, it suffices to check that Fix δ ∈ (0, 1) and set u m = δ m 1−λ .Then Using Raabe's criterion, we infer that the series ∞ m=1 u m is convergent.This finishes the proof.
Proof of Theorem 3.4: Fix z 0 ∈ C n and an open ball U := B(z 0 , r) of z 0 that satisfies the condition given in (ii).It suffices to prove that for a given ε, there exist an exceptional set A ε ⊂ U of measure less than ε and a measurable function F ε on U \A ε such that |r m −F ε | 1/m converges to 0 uniformly on U ε .To this end, observe that by (ii), there exist a constant λ ∈ (0, 1) and polynomials q m , q m satisfying the following properties: (i) r m = p m /q m , where q m = q m q m .(ii) α m := deg q m ≤ m λ for every m ≥ m 0 and q m is zero free on U .After shrinking U and making a normalization, we may achieve that q m is zero free on a fix neighborhood V of U and that (3.4) q m U = q m U = 1.
It follows that q m U ≤ 1.Then by Bernstein-Markov's inequality we obtain for every compact subset K of C n .We claim that there exists a compact non-pluripolar subset X of X such that sup To see this, set By assumption (i), we have N ≥1 X N = X.Since X is non-pluripolar, we infer that there must exist N 0 such that X N0 is non-pluripolar.Since X N0 is Borel, it must contain a non-pluripolar compact subset X .Since f is bounded on X, we obtain The claim follows by combining the above inequality with (3.5).Thus, using again Bernstein-Markov's inequality we derive for every compact subset K of C n .For m ≥ 1 we define Using (3.5), (3.6), and simple estimates, we obtain that the sequence {u m } m≥1 is uniformly bounded from above on compact sets of C n .The key step is to show that {u m } m≥1 converges uniformly to −∞ on compact sets of C n .To see this, we first observe the trivial equality It follows from (i) and (3.5) that Since X is non-pluripolar, using the same argument as in the proof of Theorem 3.1, we infer that {u m } m≥1 must converge uniformly to −∞ on compact sets of C n .In particular, for a given M > 0, there exists Thus the following estimates hold on U Next, fix m ≥ m 0 , we will use an idea from [4] to estimate the size of the subsets of U on which q m+1 q m is small.More precisely, for δ ∈ (0, 1) we set Recall that λ 2n denotes the Lebesgue measure in C n .By Corollary 4.2 in [1], we have the following estimates Here C n,r0 is a positive constant depending only on n, r 0 .So for every m ≥ 1 we get It follows that where A ε := m≥1 X m,δ .By assumption (ii) and Lemma 3.5, we can choose δ ∈ (0, 1) so small such that the right hand side of the last inequality is less than ε.
On the other hand, since q m is zero free on V if m ≥ m 0 , the function u m := 1 m log |q m | is pluriharmonic on V for every m ≥ m 0 .By the normalization (3.4) and Bernstein-Markov's inequality, we obtain that sup U u m = 0 for every m ≥ 1 and that the sequence {u m } m≥1 is uniformly bounded from above on compact sets of C n .In particular, this sequence does not converges to −∞ uniformly on compact sets of V .It follows that this sequence is also uniformly bounded from below on U .Thus we get a constant C > 0 such that (3.8) inf Hence for z ∈ U ε := U \ A ε , from (3.7) and (3.8) we deduce Hence, for M large enough such that e C −M < δ, by the triangle inequality we see that {r m } m≥1 is a Cauchy sequence on U ε .Therefore, r m converges uniformly on U ε to a measurable F ε such that lim m→∞ r m − F ε 1/m Uε = 0. We are done.
Remarks.(i) We do not know if the theorem is still true without assumption (ii).On the other hand, by tracking down the above proof, we see that the conclusion of the theorem remains valid if (ii) is omitted whereas (i) is strengthen to where γ > 1 is a constant.
(ii) The proof of the theorem also shows that for any open set D ⊂ C n on which r m is holomorphic for every m, the convergence is rapidly uniformly on compact sets i.e., for every compact subset K of D we have lim m→∞ In particular, F is holomorphic on D.
(iii) By considering r m := 1/r m , it is easy to see that the theorem still holds if assumption (ii) is written in terms of the degree of the zero sets of r m .
The last result of this section may be regarded as a boundary version of Bloom's theorem (Theorem 1.2).Theorem 3.6.Let D be a bounded domain in C n and X ⊂ ∂D be a compact subset.Let f be a bounded holomorphic function on D and {r m } m≥1 be a sequence of rational functions on C n .Suppose that the following conditions are satisfied: (i) For every x ∈ X, the point rx ∈ D for r < 1 and close enough to 1. Furthermore, if u ∈ PSH(D), u < 0 and satisfies (ii) For every x ∈ X, there exists the limit Then the following assertions hold true: For every z 0 ∈ D \ E and every affine complex subspace L of C n passing through z 0 , there exists a subsequence {r mj } j≥1 such that converges to 0 in capacity (with respect to L).Here D z0 denotes the connected component of D ∩ L that contains z 0 .
Remarks.(i) The first condition imposed on X is local, and if ∂D is C 1 -smooth at points of X then it is valid after a suitable change of coordinates.
(ii) If X is a compact subset of the unit circle with positive Lebesgue measure then X satisfies assumption (i).This can be seen as follows.Let u ∈ SH(∆), u < 0, where ∆ is the unit disk in C be such that lim We have to show that u ≡ −∞.Fix a point ξ ∈ ∆, by composing with an automorphism of ∆ we may assume that ξ = 0.Then, using the mean value inequality and Fatou's lemma we obtain We are done.
(iii) Part (b) of Theorem 3.6 has not been studied before by either Bloom or Gonchar even in the case where X is a non-pluripolar subset of D.
It is inspired by a result of Sadullaev in [10] which states that locally a holomorphic function can be rapidly approximated (in measure) if and only if its restriction to every complex line can be rapidly approximated.
(iv) The main difficulty that leads to the passage into subsequence in (b) lies in the fact that the complex subspace L may be disjoint from the non-pluripolar set X, thus a direct application of (a) is not possible then.
For the proof of the theorem, we first introduce the following notation: Let D be a bounded domain in C n and E be a subset of ∂D.Then we define the following variant of the relative extremal function Using the above terminology, we have the following lemma which exploits property (i) of the set X given in Theorem 3.6.Lemma 3.7.Let D be a bounded domain in C n and X be a subset of ∂D.Suppose that X satisfies condition (i) of Theorem 3.6.Then for every sequence {X j } j≥1 ⊂ ∂D such that X j ↑ X we have Proof: Assume that the conclusion is false.Since {ω R (z, X j , D)} j≥1 is a decreasing sequence of non-positive functions, we infer that there exists Next, we set It is clear that ϕ ∈ PSH(D), ϕ < 0, ϕ(z 0 ) > −1.Now, for a given x ∈ X we choose j x such that x ∈ X jx .Then we have This is a contradiction and the proof is thereby completed.
We also require some standard facts about compactness in the cone of plurisubharmonic functions.Proof: Assertions (a) and (b) follow from Theorem 3.2.12 in [8].Next, we apply again Theorem 3.2.12 in [8] to obtain u = (lim sup j→∞ u mj ) * everywhere on D. Hence, by Lemma 2.3 we get (c).Finally, (d) follows easily from (c), so we conclude the lemma.
The final ingredient is a sufficient condition for a sequence of measurable functions converging in capacity to 0. Lemma 3.9.Let {u m } m≥1 be a sequence of plurisubharmonic functions and {v m } m≥1 be a sequence of measurable functions defined on a bounded domain D ⊂ C n .Assume that the following conditions are satisfied: (i) {u m } m≥1 is uniformly bounded from above.
(ii) There exists a compact subset X of D such that Then the sequence {e vm } m≥1 converges to 0 in capacity.
Proof: Assume otherwise, then there exists a compact subset K of D, a subsequence {m j } and constants 0 where K j := {z ∈ K : v mj < log ε}.It follows from (ii) that u mj does not go to −∞ uniformly on X.So using Lemma 3.7 and assumption (i), we may assume, after passing to a subsequence that u mj converges in L 1 loc (D) to u ∈ PSH(D), u ≡ −∞.Next, from assumption (iii) we infer that for every M such that M + log ε > 0 there exists j M ≥ 1 such that u mj + v mj < −M, ∀ j ≥ j M , ∀ z ∈ K.So for j ≥ j M we have the following inclusion Let ω be a small neighborhood of K which is relatively compact in D. Then we have (3.9)sup According to Definition 2.1, we may choose By a version of Chern-Levine-Nirenberg's inequality (see Theorem 2.1.7 in [2]) we obtain for every j ≥ j M the following estimates Here C K,ω is a positive constant dependent only on K, ω.By letting M → ∞ and applying (3.9) we get a contradiction.The proof is complete.
Proof of Theorem 3.6: (a) After removing from X a pluripolar subset (possibly empty), we may assume that r m (z) ∈ C for every z ∈ X and m ≥ 1.Since f * is bounded on X, from assumptions (ii) and (iii) we infer that For N ≥ 1 we let It follows that X = N ≥1 X N .Since X is non-pluripolar, we deduce that there exists N 0 ≥ 1 such that X := X N0 is non-pluripolar.Now we write r m = p m /q m with q m is normalized so that q m X = 1.For m ≥ 1, we define the following plurisubharmonic functions on D We claim that the sequence {u m } m≥1 converges to −∞ uniformly on compact sets of D. For this purpose, observe that, since X is nonpluripolar, Bernstein-Walsh's inequality (2.1) implies that the sequence {v m } m≥1 is uniformly bounded from above on compact sets of C n .By the choice of X , using Bernstein-Walsh's inequality (2.1) again, we deduce that the sequence 1 m log |p m | is uniformly bounded from above on compact sets of C n as well.It follows, using easy estimates and the assumption on boundedness of f , that the sequence {u m } m≥1 is uniformly bounded from above on compact sets of D. For k, j ≥ 1 we let Let v j and v j be the restrictions to D z0 of the two sequences v mj and 1 mj log |r mj − f |.According to the results provided in (a), we have v j + v j converges uniformly to −∞ on compact sets of D z0 .So using again Lemma 3.8 we conclude that e v j converges rapidly in capacity to 0 in D z0 .We are done.

Explicit constructions of rapid convergence
The goal of this section is to provide an example of a sequence of rational functions satisfying the assumptions of Theorem 3.6.More precisely, we will construct a sequence of rational functions {r m } m≥1 with poles lying outside ∆ such that {r m } m≥1 converges rapidly pointwise to f * on some compact subset of ∂D.Here f * is the radial boundary values of a bounded holomorphic function f defined on the unit disk ∆.We begin with a general criterion which guarantees rapid convergence of certain infinite products.(i) {r m } m≥1 is locally uniformly bounded on D.
(ii) lim m→∞ ∞ j=m+1 β j 1 m = 0. (iii) There exists a non-pluripolar subset X of D such that for every x ∈ X, there exists a constant M x > 0 such that Then the sequence {r m } m≥1 converges rapidly uniformly on every compact subset of D to a holomorphic function f on D.
Proof: It follows from assumptions (i) and (iii) that the infinite product r 1 m≥1 rm+1 rm converges pointwise to a complex valued, measurable function g on X.We claim that r m converges rapidly pointwise to g on X.For this, we write f j := r j /r j−1 for j ≥ 2. Fix x ∈ X.
In view of (iii) we obtain the following estimates when m is sufficiently enough Here we use the inequality e t ≤ 1 + 2t, 0 ≤ t 1 and The claim now follows from (ii).Finally, it suffices to apply remark (ii) after the proof of Theorem 3.4 to reach the desired conclusion.
For a precise construction, we first fix a compact subset F of the unit circle ∂∆ = {z ∈ C : |z| = 1} which is of positive length but is nowhere dense in ∂∆.In particular, F satisfies condition (i) of Theorem 3.6.This can be done by taking a Cantor set C ⊂ (−1, 1) with the same property and then taking F := ∂∆ ∩ π −1 (C), where π is the orthogonal projection from ∂∆ onto the real axis.The proof proceeds through two lemmas.For the first one, we need to introduce some notation.We index (N * ) 2 = {(m, n) : m, n ≥ 1} by the graded lexicographic order, that is (m, n) ≺ (p, q) if and only if m + n < p + q, or m + n = p + q but m < p.Let ind(m, n) is the index of (m, n) in the ordered sequence.Simple computation gives for all m, n ≥ 1.
Lemma 4.3.There exists a double-indexed sequence {r mn } ⊂ (0, 1) such that the corresponding graded lexicographic sequence {s j } j≥1 satisfies the condition Proof: Fix a real number a > 1.We will prove that the sequence defined by r jk = 1 − a −j 4 −k 4 satisfies the condition.Indeed, we know that r jk = s ind(j,k) and max(j, k) ≥ ind(j,k)
Next, we estimate each term in the last expression.We see that The last estimate implies the desired limit.Proof: We write where in the last estimate we use the inequality sin t ≥ 2 π t for all 0 ≤ t ≤ π 2 .
Lemma 4.5.There exists a sequence B = {α j } j≥1 ⊂ ∆ such that (a) F is the set of accumulation points of B.
Proof: We use an argument of Colwell in [5].The set ∂∆ \ F is a countable union of disjoint open arcs.Let A be the set of end-points of these arcs.We write For each m, n ≥ 1, we define two double-indexed sequences by Notice that, for fixed m, ξ mn ↓ ϕ m and η mn ↑ ψ m .We let t mn := 1 − (1 − r mn ) ψm−ϕm 2π n < 1 − r mn where the r mn are defined in Lemma 4.3.
Let us set c mn = t mn e iξmn , d mn = t mn e iψmn .

It is easily seen that
For the last assertion, note that Let A := { 1 ᾱj : j ≥ 1}.Then the set of accumulation points of A is F .We fix a compact set K ⊂ C \ A. For j ≥ 1 and z ∈ K we have Assertion (c) now follows from Lemma 4.5(c).Finally, we observe that since the zero set of r m is exactly {α 1 , . . ., α m } and since every point of F is an accumulation point of {α m } m≥1 , the limiting function f can not be extended holomorphically through any point of F .This completes the proof.

Lemma 2 . 3 .
Let {u m } m≥1 be a sequence of plurisubharmonic functions on D. Assume that the sequence is uniformly bounded from above on compact sets of D. Let u(z) := lim sup m→∞ u m (z), z ∈ D.

( a )
The sequence |r m − f | 1/m converges in capacity to 0 on D. (b) There exists a pluripolar subset E of C n with the following property:

Lemma 3 . 8 .
Let {u m } m≥1 be a sequence of plurisubharmonic functions defined on a domain D in C n .Suppose that the sequence is uniformly bounded from above on compact subsets of D and does not converge to −∞ uniformly on some compact subset of D. Then the following assertions hold:(a) There exists a subsequence {u mj } j≥1 converging in L 1 loc (D) to a function u ∈ PSH(D), u ≡ −∞.(b) lim sup j→∞ u mj ≤ u on D. (c) lim sup j→∞ u mj = u outside a pluripolar subset of D. (d) The set {z ∈ D : lim j→∞ u mj (z) = −∞} is pluripolar.
let D z0 be the connected component of D ∩ L that contains z 0 .Choose a subsequence {v mj } j≥1 such that inf j≥1 v mj (z 0 ) > −∞.

Proposition 4 . 1 .
Let {r m } m≥1 be a sequence of rational functions, D a domain in C n , and {β m } m≥1 a sequence of positive numbers.Suppose that the following conditions are satisfied:

Proposition 4 . 2 .
There exist a countable subset A of C \ ∆ with F ⊂ A, a sequence {r m } m≥1 of rational functions on C, and a holomorphic function f : C \ A → C which is bounded on ∆ such that the following properties holds true: (a) The poles of {r m } m≥1 are included in A for every m ≥ 1.(b) {r m } m≥1 converges rapidly uniformly on compact sets of C\A to f .(c) {r m } m≥1 converges rapidly pointwise on F = A\A to f * , the radial boundary values of f .(d) f does not extend through any point of F to a holomorphic function.