I think a suitable version of the [Ham Sandwich Theorem](http://en.wikipedia.org/wiki/Ham_sandwich_theorem) for measures should imply the stronger result that it is always possible to divide the land so that each citizen in each state believes that her state received *exactly* $\frac{1}{2}$ the land value.  

See this [paper](http://www.emis.de/journals/AM/06-1/am1271.pdf) of Golasinski for multiple measure-theoretic versions of the Ham Sandwich theorem.