| The space of measurable functions on the unit interval, {dollar}Lsb0{dollar} is the classical example of an F-space. We consider the closed linear span of the Rademacher functions, {dollar}rsb{lcub}n{rcub}(x) = sgn (sin(2pi nx)),{dollar} in {dollar}Lsb0{dollar} with the topology of convergence in measure. It is shown that this subspace, called {dollar}{lcub}rm I!D{rcub},{dollar} is isomorphic to the sequence space {dollar}ellsb2.{dollar} We then show that there are relatively few composition operators on the quotient of {dollar}Lsb0{dollar} by {dollar}{lcub}rm I!D{rcub}.{dollar} As a counter point, we show that the quotient is actually transitive and even more, that the quotient is n-transitive. This last is a consequence of the construction of an operator on {dollar}Lsb0{dollar} whose kernal is exactly {dollar}{lcub}rm I!D{rcub}.{dollar}... |
