  MV-algebras can be viewed either as the Lindenbaum algebras of Lukasiewicz
infinite-valued logic, or as unit intervals of lattice-ordered abelian groups
in which a strong order unit has been fixed. The free n-generated MV-algebra
Free_n is representable as an algebra of continuous piecewise-linear functions
with integer coefficients over the unit cube [0,1]^n. The maximal spectrum of
Free_n is canonically homeomorphic to [0,1]^n, and the automorphisms of the
algebra are in 1-1 correspondence with the pwl homeomorphisms with integer
coefficients of the unit cube. In this paper we prove that the only probability
measure on [0,1]^n which is null on underdimensioned 0-sets and is invariant
under the group of all such homeomorphisms is the Lebesgue measure. From the
viewpoint of lattice-ordered abelian groups, this fact means that, in relevant
cases, fixing an automorphism-invariant strong unit implies fixing a
distinguished probability measure on the maximal spectrum. From the viewpoint
of algebraic logic, it means that the only automorphism-invariant truth
averaging process that detects pseudotrue propositions is the integral with
respect to Lebesgue measure.
