There is a "cake" which is represented by the interval $[0,1]$.
There is a non-atomic value measure $V$ defined on the cake.
I would like to define an algorithm for dividing the cake between two people, which should go roughly like this:
- Give the valuable parts of the cake to person A;
- Give the valueless parts to person B.
A "valuable" part is a part with positive value and a "valueless" part is a part with zero value. Intuitively, this is easy to understand. E.g, if the linked graph describes the value of the interval $[0,x]$, then the piece $[0.2,0.4]$ is valueless and all other pieces are valuable. But how can I define this formally? Some options which I considered are:
- The valueless part is the largest subset $X$ of the cake for which $V(X)=0$. This definition seems incorrect, since the value of e.g. $[0.2,0.4]\cup{0.6}$ is also 0, and it is larger than $[0.2,0.4]$.
- The valueless part is the subset of the cake on which the derivative of $V$ is 0: $\{x|V'(x)=0\}$. But, how can I be sure that this subset is indeed measureable?
What would be a good definition to capture this intuition?
The usual definition to make in this context is to say that the "valueless" part is the largest open set which has zero value. As you mention, there is typically no largest set which has zero value. However, there is always a largest open set of zero value: take the union of all open sets of zero value. This union will still have zero value, because it is equal to the union of countably many such open sets (by second-countability; explicitly, it is equal to the union of all intervals with rational endpoints which have zero value). The complement of this open set is a closed set called the support of the measure.