About amenability of subgroups of the group of diffeomorphisms of the   interval [0,1]
  Averaging linear functional on the space continuous functions of the group of diffeomorphisms of interval is found. Amenability of several discrete subgroups of the group of diffeomorphisms $\Diff^3([0,1])$ of interval is prove. In particular, a solution of the problem of amenability of the Thompson's group $F$ is given. 
E.T.Shavgulidze
math.GR math.FA