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 
and full classes 
. The starting – point of our consideration is the following
observation of the work Estrada – Kanwal, [3] : let 
be a series with positive terms. If for each 
with 
we have 
, 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 
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 
is a compact
submeasure. We also show that the upper Alexanders density is a compact
submeasure and the class of all sets 
with 
is a full class.
We recall the concept of asymptotic, uniform and upper Alexanders densities, further the concept of submeasures and full classes.
Let 
. Denote by 
,
the number of elements of the set 
. Then the numbers 
and 
are called the lower
and upper asymptotic density of the set 
, respectively. If
, then 
is said to be the
asymptotic density of the set 
.
Further we put 
. It can be shown that there exist 
(the lower and upper
uniform density of the set 
, cf. [1] ). It can
be easily seen that
.
We
recall the concept of a submeasure on 
. A function m : 
is said to be a
compact submeasure on 
provided that 
satisfies the following
four conditions:
, (cf. [5]).
for each 
we have 
for each 
there exists a
decomposition 
of 
such that
for each 
.
Theorem A. Paštéka 1990.
Let 
be a compact
submeasure on
and 
a series with positive terms.
If for each 
with 
we have 
, then 
.
The
upper asymptotic density 
is a compact
submeasure on 
. Therefore as a consequence of Theorem A we obtain the
following result.
Theorem B. Estrada – Kanwal 1986.
Let 
be a series with positive terms.
If for each 
with 
we have 
, then 
.
Main results
The
natural question arises whether the function 
can be replaced in Theorem B. by the function 
. The following theorem give positive answer to this
question.
Theorem 1.
Let 
be a series with positive terms.
If for each 
with 
we have 
, then 
.
Proof.
It can be easily cheeked that 
is a compact
submeasure. Especially if 
then we choose an 
such that 
. Then 
is the desired
decomposition of 
in the definition of
the compact submeasure.
In
what follows we assume 
and 
. If 
, we put 
and 
where 
and 
is the characteristic
function of A. The 
denotes upper, the 
lower Alexanders
density of the set 
. If 
, then 
is said to be the
Alexanders density of the set 
. Taking 
the function 
will mean upper
asymptotic density 
, the upper logarithmic density 
, respectively.
Theorem 2.
Let 
.
Suppose that 
.
Then 
is a compact
submeasure.
Finally, we recall the concept of a full class. A class 
is said to be a full class if the following conditions are
satisfied:
1)
N 
2)
if 
and 
, then 
3)
if 
is a series with
positive terms and 
for each set 
, then 
.
A trivial example of a full class is 
. Further the class 
of all 
with 
is a full class (see
Theorem B.) an analogous statement holds for the class 
of all 
with 
(see Theorem 1.). Note that 
according to 
. It can be shown that this inclusion is strict ( Example:
put 
).
Theorem 3.
If
sequence 
have a bounded
variation, 
, then class 
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.