  We consider the sequence of the hyperspheres $M_{n,r}$ i.e. the homogeneous
transitive spaces - of the Cartan subgroup $SDiag(n,\Bbb R)$ of the group
$SL(n,\Bbb R), n=1 ...$, and studied the normalized limit of the corresponding
sequence of the invariant measures $m_n$ on those spaces. In the case of
compact groups and homogeneous spaces, as example - for classical pairs
$(SO(n), S^{n-1}), n=1 ...$ - the limit of corresponding measures is the
classical infinite dimensional gaussian measure - this is well-known
Maxwell-Poincare lemma. Simultaneously that Gaussian measure is a unique (up to
scalar) invariant measure with respect to the action of infinite orthogonal
group $O(\infty)$. This coincidences means the asymptotic equivalence between
grand and small canonical ensembles for the series of the pairs $(SO(n),
S^{n-1})$. Our main result shows that situation for noncompact groups, for
example for the case $(SDiag(n,\Bbb R),M_{n,r})$ (the definitions see below),
is completely different: the limit of measures $m_n $ does not exist in literal
sense, and we show that only normalized logarithmic limit of the Laplace
transform of those measures does exist. In the same time there exists the
measure which is invariant measure with respect to continuous analogue of
Cartan subgroup of the group $GL(\infty)$ - this is so called infinite
dimensional Lebesgue (see \cite{V}). This difference is an evidence of the
non-equivalence between the grand and small canonical ensembles for the
noncompact case.
