I think a suitable version of the 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 of Golasinski for multiple measure-theoretic versions of the Ham Sandwich theorem.

Since it is possible that there are citizens in different states with the exact same valuation of land, we might as well solve the stronger version where each citizen in each state believes that her state received exactly $\frac{1}{2}$ the land value.

In this case, I think a suitable version of the Ham Sandwich Theorem for measures should imply the existence result under very general conditions.

See this paper of Golasinski for multiple measure-theoretic versions of the Ham Sandwich theorem.