Abstract. We construct a (non-integrable) function $f$ and two measure preserving, ergodic transformations ${\bf S}$ and ${\bf T}$ on a measure space $({\cal X}, {\cal A},\mu )$, $\mu({\cal X})=1$, in such a way that the ergodic means $\lim_{n\to\infty }{1\over n} \sum_{k=1}^n f({\bf S}^k x)$ and $\lim_{n\to\infty } {1\over n}\sum_{k=1}^n f({\bf T}^k x)$ exist for almost all $x$, they are finite constants not depending on $x$, but these constants differ when we are averaging with respect to the operators ${\bf S}$ and ${\bf T}$. This means that in the case of a non-integrable function $f$ and an ergodic transformation ${\bf T}$ the ergodic mean depends not only on the function $f$, but also on the transformation ${\bf T}$. The construction applies some probabilistic arguments.

AMS Subject Classification (1991): 47A35

Received July 26, 1995 and in revised form December 14, 1995. (Registered under 5723/2009.)