  We construct and study the one-parameter semigroup of $\sigma$-finite
measures ${\cal L}^{\theta}$, $\theta>0$, on the space of Schwartz
distributions that have an infinite-dimensional abelian group of linear
symmetries; this group is a continual analog of the classical Cartan subgroup
of diagonal positive matrices of the group $SL(n,R)$. The parameter $\theta$ is
the degree of homogeneity with respect to homotheties of the space, we prove
uniqueness theorem for measures with given degree of homogeneity, and call the
measure with degree of homogeneity equal to one the infinite-dimensional
Lebesgue measure $\cal L$. The structure of these measures is very closely
related to the so-called Poisson--Dirichlet measures $PD(\theta)$, and to the
well-known gamma process. The nontrivial properties of the Lebesgue measure are
related to the superstructure of the measure PD(1), which is called the conic
Poisson--Dirichlet measure -- $CPD$. This is the most interesting
$\sigma$-finite measure on the set of positive convergent monotonic real
series.
