What is entropy of a set?

[phoenixthoth]

What is the entropy of a set?

One of the two should be a general guidline:

# A measure of the disorder or randomness in a closed system.
# A measure of the loss of information in a transmitted message.

I've seen topological entropy (bowen) and entropy of random variables, but what about of sets?

 

[phoenixthoth]

mutual information

What I'm really getting at is the so called "mutual information" one set A has of another set B.

This is defined in information theory if A and B are random variables.

I want it if they are general sets.

I had a 'thought.' Maybe I can look at the smallest sigma algebra G containing A and B (I don't mean the intersection), and invent a nontrivial probability measure on this G. This turns A and B into events. Then the formula I've seen for mutual information is this:
I(A;B)=Log_2 (P(A&B) / (P(A)P(B))).

But what would be a nontrivial probability measure to put so that P(A), P(B) ∈ [0,1]. Also, P(G)=1. Is there some canonical nontrival P() that I can construct? How would I do this?