  We give a kind of \lq \lq approximate majorant principle\rq \rq \thinspace
result for the \lq \lq modified Selberg integral\rq \rq, say $\modSel_f(N,h)$,
of essentially bounded $f:\N \rightarrow \R$ (i.e., bounded by arbitrary small
powers); i.e., we get an upper bound, in terms of the modified Selberg integral
of a related function $F$ (with $|f\ast \mu|\ll F\ast \mu$, in the supports
intersection), getting a \lq \lq square-root cancellation\rq \rq \thinspace for
the error-terms. Here $\modSel_f(N,h)$ is the mean-square (in $N<x\le 2N$) of
the \lq \lq averaged short sum\rq \rq \thinspace of, say, $f:=g\ast \1$, minus
its expected value; i.e., ${1\over h}\sum_{m\le h}\sum_{0\le
|n-x|<m}f(n)-M_f(x,h)$, with expected value $M_f(x,h)$ (say, $\approx
h\sum_{d\le x}g(d)/d$); so, this mean-square weights, on average, the
$f-$values in (almost all, i.e. all, but $o(N)$ possible exceptions) the short
intervals $[x-h,x+h]$, with mild restrictions on \thinspace $h$ \thinspace
(say, \thinspace $h\to \infty$ \thinspace and \thinspace $h=o(N)$, when
\thinspace $N\to \infty$).
