Is it true that for a given nonnegative number, there exists a measuretheoretical entropy value (supremum of entropies of all partitions under a measurepreserving transformation) that equals this number?

There is also a paper of Grillenberger, where You can find the construction http://link.springer.com/article/10.1007%2FBF00537161?LI=true 


There are two funny approaches:


