A self-similar measure is a probability measure invariant under a weighted finite iterated
function system of contracting similarities (Hutchinson
1981). On the real line, let 
, where 
, and choose positive
weights 
summing to 1. Its defining equation is
 |
where the pushforward satisfies 
. The measure
is also the distribution of the random
coding limit 
,
where the symbols 
are independent random variables with
probabilities 
.
Let 
be the random composed map. Using natural
logarithms in the Shannon entropy 
, define
 |
and let 
.
The Shannon entropy counts distinct composed maps, including their combined probabilities
when different addresses coincide.
