Subadditive measures on N and the convergence of series with positive terms
T. Šalát – T. Visnyai
In this my talk we shall study the convergence of series with positive terms in connection
with a submeasure on P N  and full classes S  P N  . The starting – point of our
consideration is the following observation of the work Estrada – Kanwal, [3] : let

a
k 1
series with positive terms. If for each A  N with d  A  0 we have
a
kA

a
k 1
k
k
k
be a
  , then
  . This result is formulated in a stronger form in the work [6]. There it is shown,
that in last result the function d can be replaced by the function u (uniform density).
Consequently, if series with positive terms convergences for each n  A where A is set of
null uniform density then it convergences on N. In the proof of this assertion we use the fact
that uniform density of a set A  N is a compact submeasure. We also show that the upper
Alexanders density is a compact submeasure and the class of all sets A  N with h A  0 is
a full class.
We recall the concept of asymptotic, uniform and upper Alexanders densities, further the
concept of submeasures and full classes.
Let A  N . Denote by Am  1, m  n, m  0 , n  N the number of elements of the set
A  m  1, m  n . Then the numbers d  A  lim inf
n 


A1, n 
A1, n 
and d  A  lim sup
are
n


n
n
called the lower and upper asymptotic density of the set
A  N , respectively.

If d  A  d  A  d  A , then d  A is said to be the asymptotic density of the set A  N .

Further we put  n  lim inf Am  1, m  n,  n  lim sup Am  1, m  n . It can be shown
m
that there exist u  A  lim

n 
m
n
n

, u  A  lim
n 
n
n
(the lower and upper uniform density of the
set A  N , cf. [1] ). It can be easily seen that
1


u A  d  A  d  A  u A .


1
We recall the concept of a submeasure on P N   2 N . A function m : P N   0,  
is said to be a compact submeasure on P N  provided that m satisfies the following four
conditions:
A  B  m A  mB
i 
ii 
iii 
iv 
m A  B  m A  mB , (cf. [5]).
for each n  N we have mn  0
for each   0 there exists a decomposition N  A1  A2  ...  As of N such that
mA j    for each j  1,2,...s .
Theorem A. Paštéka 1990.
Let m be a compact submeasure on P N  and

a
k 1
If for each A  N with m A  0 we have
k
a series with positive terms.
 ak   , then
kA

a
k 1
k
  .

The upper asymptotic density d is a compact submeasure on P N  . Therefore as a
consequence of Theorem A we obtain the following result.
Theorem B. Estrada – Kanwal 1986.

Let
a
k 1
k
be a series with positive terms.
If for each A  N with d  A  0 we have
a
kA
k
  , then

a
k 1
k
  .
Main results
The natural question arises whether the function d can be replaced in Theorem B. by the
function u . The following theorem give positive answer to this question.
Theorem 1.

Let
a
k 1
k
be a series with positive terms.
If for each A  N with u A  0 we have
 ak   , then
kA
2

a
k 1
k
  .
Proof.

It can be easily cheeked that u is a compact submeasure. Especially if   0 then we choose
an m  N such that


m 1   . Then N  0  1  ...  m  1 is the desired decomposition
of N in the definition of the compact submeasure. 

c
In what follows we assume cn  0, n  N and

h A  lim sup
n 
1
sn
n
 ck  A k  and h A  lim inf
n

k 1
and  A is the characteristic function of A.
1
sn
n 1
n
  . If A  N ,
n
 c  k  where
k
k 1
we put
s n  c1  ...  cn , n  N
A

The h A denotes upper, the h A lower


Alexanders density of the set A  N . If h A  h A  h A , then h A is said to be the

Alexanders density of the set A  N . Taking c n  1, c n 


1
, (n  1,2,...) the function h
n

will mean upper asymptotic density d , the upper logarithmic density  , respectively.
Theorem 2.

Let c n  0 n  1,2,...,
c
n 1
n
  .

Suppose that Var c   c n  c n 1   .
n 1

Then h is a compact submeasure.
Finally, we recall the concept of a full class. A class S  P N  is said to be a full class if
the following conditions are satisfied:
1) N   S
2) if A  S and B  A , then B  S

3) if
t
n 1

t
n 1
n
n
is a series with positive terms and
t
nA
  .
3
n
  for each set A  S , then
A trivial example of a full class is P N  . Further the class  d0 of all A  N with d  A  0
is a full class (see Theorem B.) an analogous statement holds for the class  u0 of all A  N
with u A  0 (see Theorem 1.). Note that  u0   d0 according to 1 . It can be shown that this

inclusion is strict ( Example: put A   n!  1, n!  2,..., n!  n ).
n 1
Theorem 3.
If sequence
cn 1
have a bounded variation, c n  0 n  1,2,...,

c
n 1
n
  , then class
 h0  A  N , h A  0 is a full.
Rerences.
[1] Alexander,R. : Density and multiplicative structure of sets of integers. Acta Arithm. XII.
(1967), 321 – 329.
[2] Mačaj, M. – Mišík, L. – Šalát, T. – Tomanová, J.: On class of densities of sets of positive
integers. AMUC LXXII. (2003), 213 – 221.
[3] Estrada, R. – Kanwal, R.P.: Series that converge on sets of null density, Proc. Amer. Math.
Soc. 97 (1986), 682 – 686.
[4] Paštéka, M.: Convergence of series and submeasures of the set of positive integers,
Math. Slov. 40 (1990), 273 – 278.
[5] Sember, J.J.- Freedman, A. R.: On summing sequences of 0’s and 1’s, Rockey Mountain
J. Math. 11 (1981), 419 – 425.
[6] Šalát, T. – Visnyai, T.: Subadditive measures on N and the convergence of series with
positive terms, Acta Mathematica 6 (2003), 43 – 52.
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