  Our aim is to find the irreducibility criteria for the Koopman
representation, when the group acts on some space with a measure (Conjecture
1.5). Some general necessary conditions of the irreducibility of this
representation are established. In the particular case of the group ${\rm
GL}_0(2\infty,{\mathbb R})$ $= \varinjlim_{n}{\rm GL}(2n-1,{\mathbb R})$, the
inductive limit of the general linear groups we prove that these conditions are
also the necessary ones. The corresponding measure is infinite tensor products
of one-dimensional arbitrary Gaussian non-centered measures. The corresponding
$G$-space $X_m$ is a subspace of the space ${\rm Mat}(2\infty,{\mathbb R})$ of
infinite in both directions real matrices. In fact, $X_m$ is a collection of
$m$ infinite in both directions rows. This result was announced in [20]. We
give the proof only for $m\leq 2$. The general case will be studied later.
