(Uniqueness of elementary measure). Let . Let be a map from the collection of elementary subsets of to the non-negative reals that obeys the non-negativity, finite additivity and translation invariance properties. Show that there exists a constant such that for all elementary sets . In particular, if we impose the additional normalisation , then . (Hint: Set , then compute for any positive integer .)
For a box we know that and we are given that so provided we can always write with .
Consider two disjoint boxes and with and let
As and both obey the finite additivity property we have
This will only be true for all boxes and if
and so for all boxes we have a constant such that
If we consider the box with a positive integer we have
and considering the limit as we obtain the result that .
Using these results and the results of the previous two exercises we can see that for any elementary set
If we normalise so that then and we have .