����������� ������ 51 (1999), 1–8

UDK 517.521

���������� ������ ��� research paper

BAIRE’S SPACE OF PERMUTATIONS OF N AND REARRANGEMENTS OF SERIES

Tibor Šalát

Abstract. In the first part of the paper we investigate the structure of the space (S, d) of all sequences of positive integers with Baire’s metric. In the second part we study properties of the space (E, d) of all permutations of N in connection with rearrangements of non-absolutely convergent series.

0. Introduction

There are several papers investigating rearrangements of non-absolutely convergent series from the point of view of permutations of the set N considered as points of a metric space (cf. [1], [3], [4], [5], [7], [8], [9], [10]). The mentioned metric space is endowed with the Fréchet’s metric d_{1} in [1], [3], [4], [7], [9], [10] and with Baire’s metric d in [5]. Let us remark that these two metrics are equivalent on the set S (and also on E) of all sequences of positive integers (of all permutations of N) and therefore many properties of (S, d_{1}) can be transfered from (S, d_{1}) to (S, d) (or from (E, d_{1}) to (E, d)), and conversely.

Several of our considerations will be based on the following classical Riemann’s theorem on rearrangements of series with real terms (cf. [5]).

Theorem A. Let ∑_{k=1}^{∞} a_{k} be a non-absolutely convergent series with real terms, let −∞ ⩽ t_{1} ⩽ t_{2} ⩽ +∞. Then there exists a permutation x = (x_{j})_{1}^{∞} ∈ E such that lim inf_{n→∞} S_{n}(x) = t_{1}, lim sup_{n→∞} S_{n}(x) = t_{2}, where S_{n}(x) = ∑_{j=1}^{n} a_{x_{j}} (n = 1, 2,…).

Particularly, for every r ∈ R there exists a permutation x = (x_{j})_{1}^{∞} ∈ E such that

∑_{j=1}^{∞} a_{x_{j}} = r.  (1)

AMS Subject Classification: 40 A 05

Keywords and phrases: Rearrangements of series, Baire’s space, set of first category, residual set, σ-porosity, strong porosity

The set of all permutations (of N) x = (x_{j})_{1}^{∞} with (1) will be denoted by P_{r} (in agreement with [5]). The set P_{r} depends obviously on the sequence a_{1}, a_{2}, … and therefore we shall write in detail P_{r} = P_{r}(a_{1}, a_{2},…). Put

E_{0} = E_{0}(a_{1}, a_{2},…) = ⋃_{r∈R} P_{r}(a_{1}, a_{2},…).

Although the sets E_{0}(a_{1}, a_{2},…), P_{r}(a_{1}, a_{2},…) depend on a_{1}, a_{2}, …, they have several common properties for all a_{1}, a_{2}, … such that ∑_{k=1}^{∞} a_{k} is a non-absolutely convergent series.

Recall the concept of metrics d_{1} and d (on S or E). Let x = (x_{j})_{1}^{∞} ∈ S, y = (y_{j})_{1}^{∞} ∈ S. Then we put

d_{1}(x, y) = ∑_{k=1}^{∞} \frac{1}{2^{k}} \frac{|x_{k} − y_{k}|}{1 + |x_{k} − y_{k}|}.

If x = y, then d(x, y) = 0 and if x ≠ y, then d(x, y) = 1/m, where m = min{ j : x_{j} ≠ x_{i} } (cf. [1], [2], p. 185).

We shall use the concept of porosity of sets (cf. [12], [13]). Let (Y, ρ) be a metric space, y ∈ Y and r > 0. Then by K(y, r) denote the ball with centre y and radius r, i.e. K(y, r) = { x ∈ Y : ρ(x, y) < r }. Let M ⊆ Y. Put

γ(y, r, M) = sup{ t > 0 : (∃z ∈ Y) [K(z, t) ⊆ K(y, r)] ∧ [K(z, t) ∩ M = ∅] },

\overline{p}(y, M) = lim sup_{r→0+} γ(y, r, M)/r, \underline{p}(y, M) = lim inf_{r→0+} γ(y, r, M)/r and if \overline{p}(y, M) = \underline{p}(y, M), then set

p(y, M) = \overline{p}(y, M) = \underline{p}(y, M) = lim_{r→0+} \frac{γ(y, r, M)}{r}.

Obviously, each of the numbers \overline{p}(y, M), \underline{p}(y, M), p(y, M) belongs to the interval [0, 1].

A set M ⊆ Y is said to be porous (c-porous) at y, provided that \overline{p}(y, M) > 0 (\overline{p}(y, M) ⩾ c > 0). A set M ⊆ Y is said to be σ-porous (σ-c-porous) at y provided that M = ⋃_{n=1}^{∞} M_{n}, M_{n} (n = 1, 2,…) being porous (c-porous) at y.

Let Y_{0} ⊆ Y. A set M ⊆ Y is said to be porous, c-porous, σ-porous and σc-porous in the set Y_{0} if M is porous, c-porous, σ-porous and σ-c-porous at every point y ∈ Y_{0}, respectively.

If a set M is c-porous and σ-c-porous at y, then obviously it is porous and σ-porous at y, respectively.

Every porous set M in Y ia a nowhere dense set in Y and therefore every σ-porous set M in Y is a set of the first Baire category in Y. The converse is not true already in R (cf. [11]).

From the definition of numbers \overline{p}(y, M), \underline{p}(y, M) we get at once: If M_{1} ⊆ M_{2} ⊆ Y, then for each y ∈ Y we have \overline{p}(y, M_{1}) ⩾ \overline{p}(y, M_{2}), \underline{p}(y, M_{1}) ⩾ \underline{p}(y, M_{2}).

Let Y be a metric space again. A set M ⊆ Y is said to be very porous at y ∈ Y if \underline{p}(y, M) > 0 and very strongly porous at y if p(y, M) = 1 (cf. [1], p. 327). The set M is said to be very (strongly) porous in Y_{0} ⊆ Y if it is very (strongly) porous at each point y ∈ Y_{0}.

Obviously, if M is very porous at y, it is porous at y, as well. Analogously, if M is very strongly porous at y, it is 1-porous at y.

Further, a set M ⊆ Y is said to be uniformly very porous (in Y_{1} ⊆ Y) provided that there is a c > 0 such that for each y ∈ Y_{1} we have \underline{p}(y, M) ⩾ c > 0 (cf. [13], p. 327).

In agreement with the previous terminology and in analogy with the notion of σ-porosity we introduce the following concept of uniformly very σ-porous sets.

Definition. 1) A set M ⊆ Y is said to be uniformly very σ-porous (in Y_{0} ⊆ Y) provided that M = ⋃_{n=1}^{∞} M_{n} and there is a c > 0 such that for each y ∈ Y_{0} we have \underline{p}(y, M_{n}) ⩾ c > 0 (n = 1, 2,…).

2) A set M ⊆ Y is said to be uniformly very strongly σ-porous (in Y_{0} ⊆ Y) provided that M = ⋃_{n=1}^{∞} M_{n} and for each y ∈ Y_{0} we have p(y, M_{n}) = 1 (n = 1, 2,…).

If A ⊆ Y, then by CA we denote the complement of A, CA = Y \ A.

In the first part of the paper we shall investigate the structure of the space (S, d). In the second part we shall study properties of the space (E, d) in connection with rearrangements of non-absolutely convergent series.

1. Structure of the space (S, d)

We shall study the structure of (S, d) from the point of view of its subset E and closure \overline{E} = E_{1} of E in S.

It is well-known (cf. [1], [5], [9], [10]) that the set E is not closed in S. Its closure E_{1} = \overline{E} equals to the set of all x = (x_{j})_{1}^{∞} ∈ S containing every positive integer at most once (equivalently: E_{1} consists of all one-to-one sequences of positive integers).

The metric space (E, d) (subspace of (S, d)) is of the second category at each of its points (cf. [1], [4], [5], [9], [10]). We recall the following well-known result (cf. [1], [4], [5], [7], [9], [10]):

Theorem B. Let ∑_{k=1}^{∞} a_{k} be a non-absolutely convergent series. Denote H = H(a_{1}, a_{2},…) the set of all x = (x_{j})_{1}^{∞} ∈ E such that lim inf_{n→∞} S_{n}(x) = −∞, lim sup_{n→∞} S_{n}(x) = +∞ (S_{n}(x) = ∑_{j=1}^{n} a_{x_{j}}, n = 1, 2,…). Then the set H is a residual set in E (i.e. E \ H is a set of the first Baire category in E).

This result has been strengthened in [3], where it is proved that the set of all x = (x_{j})_{1}^{∞} ∈ E with (S_{n}(x))′_{n} = [−∞, ∞] is residual in E ((S_{n}(x))′_{n} denotes the set of all limit points of the sequence (S_{n}(x))_{n=1}^{∞}).

It is well-known that E is a G_{δ}-set in S (cf. [1]). We shall complete this result in the following proposition and for the completeness we shall give also the proof of the mentioned result from [1] (see (i) in the following proposition).

Proposition 1.1. The set E ⊂ S has the following properties:

(i) The set E belongs exactly to the first Borel class, it is a G_{δ}-set in S.

(ii) The set E is dense in itself.

(iii) The set E is nowhere dense in S.

Proof. (i) Put (as in [1]) G(n, k) = {x = (x_{j})_{1}^{∞} ∈ S : x_{n} = k }. If x ∈ G(n, k) then it is easy to see that K(x, 1/n) ⊆ G(n, k), so the set G(n, k) is open in S. But then the set

E_{2} = ⋂_{k=1}^{∞} ⋃_{n=1}^{∞} G(n, k)  (2)

is a G_{δ}-set in S. Further we have obviously

E = E_{1} ∩ E_{2},  (2′)

E_{1} being closed in S. Hence by (2′) the set E is a G_{δ}-set in S.

Further, E is neither closed in S (cf. [1], [5]), nor open in S (see (iii)), so the set E belongs exactly to the first Borel class.

(ii) Let y ∈ E, ε > 0. Choose s ∈ N such that 1/s < ε. Define z = (z_{j})_{1}^{∞} as follows: Put z_{j} = y_{j} (j = 1, 2,…, s), z_{s+1} = y_{s+1} + 1 and construct the one-to-one sequence z_{s+2}, z_{s+3}, … containing all positive integers different from z_{1}, …, z_{s}, z_{s+1}. Then z = (z_{j})_{1}^{∞} ∈ E, z ∈ K(y, ε) and z ≠ y.

(iii) Let y ∈ S, ε > 0. We show that there exists a ball K_{0} ⊆ K(y, ε) such that K_{0} ∩ E = ∅. Then on account of a well-known criterion of nowhere density (cf. [6], p. 37), the assertion follows.

Choose s ∈ N such that 1/s < ε. Put z_{j} = y_{j} (j = 1, 2,…, s), z_{s+1} = z_{s} and z_{s+k} = 1 (k = 2, 3,…). Then z = (z_{j})_{1}^{∞} ∈ K(y, ε), K(z, 1/(s + 2)) ⊆ K(y, ε) and K(z, 1/(s + 2)) ∩ E = ∅.

The nowhere density of the set E in S follows also from the following result.

Theorem 1.1. The set E_{1} = \overline{E} is very strongly porous in S.

Corollary. a) The set E_{1} is nowhere dense in S.

b) The set E is nowhere dense in S.

Proof of Theorem 1.1. Let q = (q_{j})_{1}^{∞} ∈ S, 0 < ε < 1. Choose s ∈ N such that 1/s ⩽ ε < 1/(s − 1) (s ⩾ 2). Construct y = (y_{j})_{1}^{∞} ∈ S in this way: Put y_{i} = q_{i} (i = 1, 2,…, s), y_{s+1} = q_{s}, y_{s+j} = 1 (j = 2, 3,…). Then y ∈ K(q, 1/s) and K(y, 1/(s + 1)) ⊆ K(q, 1/s) ⊆ K(q, ε). If z = (z_{j})_{1}^{∞} ∈ K(y, 1/(s + 1)) then z_{s} = z_{s+1}, so K(y, 1/(s + 1)) ∩ E_{1} = ∅. But then by the definition of γ(q, ε, E_{1}) we get

\frac{γ(q, ε, E_{1})}{ε} ⩾ \frac{s − 1}{s + 1}.

If ε → 0+, the s → ∞ and so we get

\underline{p}(q, E_{1}) = lim inf_{ε→0+} \frac{γ(q, ε, E_{1})}{ε} = 1.

In the representation E = E_{1} ∩ E_{2} of the set E (see (2′)) the set E_{1} is very strongly porous in S (hence it is a nowhere dense set). We now show that E_{2} is a residual set in S. This follows from the following result.

Theorem 1.2. The set CE_{2} = S \ E_{2} is uniformly very strongly σ-porous in S.

Proof. By (2) we have E_{2} = ⋂_{k=1}^{∞} B(k), where B(k) = ⋃_{n=1}^{∞} G(n, k). Using de Morgan’s rule we get

CE_{2} = ⋃_{k=1}^{∞} CB(k),  (3)

where CB(k) = ⋂_{n=1}^{∞}{ x = (x_{j})_{1}^{∞} ∈ S : x_{n} ≠ k }.

We show that p(y, CB(k)) = 1 (k = 1, 2,…) for each y ∈ S. Then (3) gives the assertion.

Let k be fixed, y ∈ S, 0 < ε < 1. Choose an s ∈ N such that 1/s ⩽ ε < 1/(s − 1). Put z_{j} = y_{j} (j = 1, 2,…, s) and

z_{s+1} = k,  (4)

z_{s+j} = 1 (j = 2, 3,…). Then K(z, 1/(s + 1)) ⊆ K(y, 1/s) ⊆ K(y, ε) and by (4) we have K(z, 1/(s + 1)) ∩ CB(k) = ∅. Hence

\frac{γ(y, ε, CB(k))}{ε} ⩾ \frac{s − 1}{s + 1}.

If ε → 0+, then s → ∞ and so we get p(y, CB(k)) = 1.

2. Structure of the space (E, d) and rearrangements of series

If ∑_{k=1}^{∞} a_{k} is a non-absolutely convergent series then by Theorem B the set E_{0} = E_{0}(a_{1}, a_{2},…) is a set of the first category in E. We shall complete this result by showing that E_{0} is a σ-1-porous set at points of a large subset of S.

In the first place we shall investigate to which Borel classes the set E_{0}(a_{1}, a_{2},…) and the sets

H^{+} = H^{+}(a_{1}, a_{2},…) = { x ∈ E : lim sup_{n→∞} S_{n}(x) = +∞ },
H^{−} = H^{−}(a_{1}, a_{2},…) = { x ∈ E : lim inf_{n→∞} S_{n}(x) = −∞ }

belong.

Theorem 2.1. For each non-absolutely convergent series ∑_{k=1}^{∞} a_{k} the set E_{0} = E_{0}(a_{1}, a_{2},…) is an F_{σδ}-ste in E.

Remark 2.1. The foregoing theorem does not state exactly the Borel class of the set E_{0}. According to a result from [10] the set E_{0} is dense and simultaneously a boundary set in E. From this it is easy to see that E_{0} cannot belong to the zero Borel class. The set E_{0} cannot be a G_{δ}-set in E, since in the opposite case it would be residual in S (cf. [6], p. 49). But this contradicts Theorem B. One can conjecture that E_{0} is an F_{σ}-set in S, but I am not able to prove or disprove this conjecture.

Proof of Theorem 2.1. By the definition of the set E_{0} and Cuachy’s test for convergence a permutation x = (x_{j})_{1}^{∞} ∈ E belongs to E_{0} if and only if the following holds:

(∀k)(∃m)(∀j) |S_{m+j}(x) − S_{m}(x)| ⩽ \frac{1}{k} .

From this we get

E_{0} = ⋂_{k=1}^{∞} ⋃_{m=1}^{∞} ⋂_{j=1}^{∞}{ x ∈ E : |S_{m+j}(x) − S_{m}(x)| ⩽ 1/k }.  (5)

Consider that by fixed m and j the function S_{m+j}(x) − S_{m}(x) = ∑_{n=m+1}^{m+j} a_{x_{n}} is constant on every ball K(y, 1/(m + j)), y ∈ S. Hence it is continuous on S. The assertion follows from (5).

It is proved in [9] that the set B = B(a_{1}, a_{2},…) = E \ (H^{+} ∪ H^{−}) belongs to the first Borel class and is an F_{σ}-set in S. We show that this result is exact.

Theorem 2.2 If ∑_{k=1}^{∞} a_{k} is a non-absolutely convergent series, then the sets H^{+}(a_{1}, a_{2},…), H^{−}(a_{1}, a_{2},…) and B(a_{1}, a_{2},…) belong exactly to the first Borel class, the sets H^{+}, H^{−} are G_{δ}-sets and B is an F_{σ}-set in S.

Corollary. The set H^{∗} = H^{+} ∪ H^{−} belongs exactly to the first Borel class and it is a G_{δ}-set in S.

Proof of Theorem 2.2. By using Theorem A it is easy to see that each of the sets H^{+}, H^{−}, B is dense and simultaneously boundary in S. From this we see that none of these sets belongs to zero Borel class. Further

H^{+} = ⋂_{m=1}^{∞} ⋃_{n=1}^{∞}{ x ∈ E : S_{n}(x) > m }.  (6)

We show that the set A(n, m) = { x ∈ E : S_{n}(x) > m } is open in S. Indeed, if x ∈ A(n, m) then K(x, 1/n) ⊆ A(n, m). But then from (6) we get that H^{+} belongs exactly to the first Borel class and it is a G_{δ}-set in E.

Similarly it can be shown that H^{−} belongs exactly to the first Borel class and it is a G_{δ}-set in E. The statement related to B follows from B = E \ H^{∗}, where H^{∗} = H^{+} ∪ H^{−}.

Put B_{1} = B_{1}(a_{1}, a_{2},…) = { x ∈ E : (S_{n}(x))_{1}^{∞} is bounded from above }, B_{2} = B_{2}(a_{1}, a_{2},…) = { x ∈ E : (S_{n}(x))_{1}^{∞} is bounded from below }. Hence B_{1} = S \ H^{+}, B_{2} = S \ H^{−}. Then B = B_{1} ∩ B_{2} and E_{0} ⊆ B. We shall investigate the porosity of sets B_{1}, B_{2}, B, E_{0}.

Lemma 2.1. Let ∑_{k=1}^{∞} a_{k} be an arbitrary non-absolutely convergent series. Then the set B_{1} = B_{1}(a_{1}, a_{2},…) is σ-1-porous in the set H^{+} = H^{+}(a_{1}, a_{2},…).

Corollary. The sets B, E_{0} are σ-1-porous in H^{+}.

Proof of Lemma 2.1. Observe that

B_{1} = ⋃_{m=1}^{∞} D(m),  (7)

where D(m) = ⋂_{j=1}^{∞}{x ∈ E : S_{j}(x) ⩽ m } (m = 1, 2,…).

Let y = (y_{j})_{1}^{∞} ∈ H^{+}. Then by definition of H^{+} there exists a sequence v_{1} < v_{2} < ⋯ < v_{k} < ⋯ of positive integers such that lim_{k→∞} S_{v_{k}}(y) = +∞. Construct the ball K(y, 1/v_{k}). Then for all sufficiently large k’s (say for k > k_{0}) we have K(y, 1/v_{k}) ∩ D(m) = ∅, thus γ(y, v_{k}^{−1}, D(m)) = 1/v_{k} (for k > k_{0}). Since v_{k}^{−1} → 0 (k → ∞), we get v_{k}γ(y, v_{k}^{−1}, D(m)) = 1. Thus \overline{p}(y, D(m)) = 1 (m = 1, 2,…). Lemma 2 follows from (7).

Similarly the following lemma can be proved.

Lemma 2.2. Let ∑_{k=1}^{∞} a_{k} be a non-absolutely convergent series. Then the set B_{2}(a_{1}, a_{2},…) is σ-1-porous in the set H^{−}(a_{1}, a_{2},…).

On account of Theorem B, using Lemma 2.1 and Lemma 2.2 we obtain the following result.

Theorem 2.3. Let ∑_{k=1}^{∞} a_{k} be an arbitrary non-absolutely convergent series. Then each of the sets B(a_{1}, a_{2},…), E_{0}(a_{1}, a_{2},…) is σ-1-porous in the residual set H^{+}(a_{1}, a_{2},…) ∩ H^{−}(a_{1}, a_{2},…).

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(received 11.06.1996.)

Comenius University, Mlynská dolina, 842 15 Bratislava, Slovakia