  We show that the order on probability measures, inherited from the dominance
order on the Young diagrams, is preserved under natural maps reducing the
number of boxes in a diagram by $1$. As a corollary we give a new proof of the
Thoma theorem on the structure of characters of the infinite symmetric group.
  We present several conjectures generalizing our result. One of them (if it is
true) would imply the Kerov's conjecture on the classification of all
homomorphisms from the algebra of symmetric functions into $\mathbb R$ which
are non-negative on Hall--Littlewood polynomials.

Combinatorics;Probability;Representation Theory