ON THE DUGUÉ PROBLEM WITH A SOLUTION IN THE SET OF SIGNEDMEASURES
Marek T. Malinowski Jolanta K. Misiewicz
Abstract: There are two methods of obtaining symmetric probability measure on a base of an
arbitrary probability measure corresponding to the random variable The first relies on
considering distribution of where is an independent copy of In the
language of measures we have then where In the
second method we consider the mean of two measures and In the paper we want to
present some known and new results on characterizing such measures for which both
methods coincide, i.e. measures for which
In
the literature one can find also the following generalization of this question: for fixed
what is the characterization of such pairs of distributions and for
which
This
problem was posed by Dugué in 1939 and it was extensively studied since then. However, the
full characterization has not been found yet. In the paper we show some constructions of the
Dugué question with the properties of simple fractions classes of characteristic functions. We
give also a collection of new solutions and an example of three measures and such
that
In
the last section we give also some solutions in the set of signed -finite measures. The
authors would like to express their gratitude to Professor D. Szynal for his interesting
questions and discussions.
corresponding to the random variable 
The first relies on
considering distribution of 
where 
is an independent copy of 
In the
language of measures we have then 
where 
In the
second method we consider the mean of two measures 
and 
In the paper we want to
present some known and new results on characterizing such measures 
for which both
methods coincide, i.e. measures for which
![p (- (0,1]](files/22.2/HTML/22.2.10.abs11x.png)
what is the characterization of such pairs of distributions 
and 
for
which
and 
such
that
-finite measures. The
authors would like to express their gratitude to Professor D. Szynal for his interesting
questions and discussions.