This reminds me of some Information Theory:

[Here](http://planetmath.org/entropyofapartition) we have a definition of Shannon entropy of a *partition* (more developed also [here](http://www.cambridge.org/9780521883894)). The book shows how the finer a partition is, the higher the resulting entropy.

And in slide 7 [here](http://math.ucr.edu/home/baez/networks_oxford/networks_entropy.pdf) Shannon entropy of a finite probability measure \\(p\\) is interpreted as

> How much information you learn, on average, when someone tells you an element \\(x \in X\\), if all youâ€™d known was that it was

randomly distributed according to \\(p\\).

So this would quantify how much more you learn by moving to finer partitions.

I take that [this](https://math.stackexchange.com/questions/381986/prove-that-it-is-a-random-variable-iff-it-is-constant-on-each-partition) MO question helps in viewing real functions on partition blocks as random variables. Problems may arise in infinite sample spaces though.

[Here](http://planetmath.org/entropyofapartition) we have a definition of Shannon entropy of a *partition* (more developed also [here](http://www.cambridge.org/9780521883894)). The book shows how the finer a partition is, the higher the resulting entropy.

And in slide 7 [here](http://math.ucr.edu/home/baez/networks_oxford/networks_entropy.pdf) Shannon entropy of a finite probability measure \\(p\\) is interpreted as

> How much information you learn, on average, when someone tells you an element \\(x \in X\\), if all youâ€™d known was that it was

randomly distributed according to \\(p\\).

So this would quantify how much more you learn by moving to finer partitions.

I take that [this](https://math.stackexchange.com/questions/381986/prove-that-it-is-a-random-variable-iff-it-is-constant-on-each-partition) MO question helps in viewing real functions on partition blocks as random variables. Problems may arise in infinite sample spaces though.