Probability and range partition in this question we have a fixed partition and we want to partition the range to obtain a three subsets with the condition below.

 A: The simplest approach is for person $1$ to produce three parts which have equal probability measure under $\rho_1$.
Person $2$ then takes those two parts with largest and second largest probability measure under $\rho_2$ and moves a bit from the former to the latter so they then have equal probability measure under $\rho_2$, each at least a third of the total.
Person $3$ can then choose one of the three parts with probability measure under $\rho_3$ of at least a third of the total. If this is one of those constructed by person $2$ then the other constructed by person $2$ has probability measure under $\rho_2$ at least a third of the total and the remaining third part has probability measure under $\rho_1$ equal to a third of the total; if person $3$ chooses the part untouched by person $2$, then one of the parts adjusted by person $2$ has has probability measure under $\rho_1$ at least a third of the total and the remaining part has probability measure under $\rho_2$ at least a third of the total.
This works but can create envy, in the sense that, although everyone gets parts which they  measure as at least a third of the total, some might think others have done even better than them.  The Selfridge–Conway discrete procedure provides an envy-free solution.  The question is usually posed as being about cakes, but the key point is that the cake is continuously divisible - hence the probability density functions used here.
