For some compact abelian groups X X (e.g. T n T^n , n ⩾ 2 n \geqslant 2 , and ∏ n = 1 ∞ Z 2 \prod \nolimits _{n = 1}^\infty {{Z_2}} ), the group G G of topological automorphisms of X X has the Haar integral as the unique G G -invariant mean on L ∞ ( X , λ X ) {L_\infty }(X,{\lambda _X}) . This gives a new characterization of Lebesgue measure on the bounded Lebesgue measurable subsets β \beta of R n {R^n} , n ⩾ 3 n \geqslant 3 ; it is the unique normalized positive finitely-additive measure on β \beta which is invariant under isometries and the transformation of R n : ( x 1 , … , x n ) ↦ ( x 1 + x 2 , x 2 , … , x n ) {R^n}:({x_1}, \ldots ,{x_n}) \mapsto ({x_1} + {x_2},{x_2}, \ldots ,{x_n}) . Other examples of, as well as necessary and sufficient conditions for, the uniqueness of a mean on L ∞ ( X , β , p ) {L_\infty }(X,\beta ,p) , which is invariant by some group of measure-preserving transformations of the probability space ( X , β , p ) (X,\beta ,p) , are described.