  The validity of the strong law of large numbers for multiple sums $S_n$ of
independent identically distributed random variables $Z_k$, $k\leq n$, with
$r$-dimensional indices is equivalent to the integrability of
$|Z|(\log^+|Z|)^{r-1}$, where $Z$ is the typical summand. We consider the
strong law of large numbers for more general normalisations, without assuming
that the summands $Z_k$ are identically distributed, and prove a multiple sum
generalisation of the Brunk--Prohorov strong law of large numbers. In the case
of identical finite moments of irder $2q$ with integer $q\geq1$, we show that
the strong law of large numbers holds with the normalisation $\|n_1\cdots
n_r\|^{1/2}(\log n_1\cdots\log n_r)^{1/(2q)+\varepsilon}$ for any
$\varepsilon>0$. The obtained results are also formulated in the setting of
ergodic theorems for random measures, in particular those generated by marked
point processes.
