Confusion about what's a measure? 
What's a distribution?
Simple but important! 
For bitstrings x: 

M measure:
M(empty string)=1
M(x) = M(x0)+M(x1) nonnegative for all finite x.

P probability distribution:
Sum_x P(x) = 1; P(x) nonnegative


M semimeasure - replace "=" by ">=":
M(x) >= M(x0)+M(x1) 

P semidistribution - replace:
Sum_x P(x) <= 1



1. Distribution: E.g., integers n: P(n) = 6/(Pi n^2)
2. Semidistribution: m(x) = probability of guessing a halting program 
   for x (BTW, Hal, this was first published by Levin in 1974, not by 
   Chaitin in 1975)
3. Measure: E.g., each x of size n gets weight 2^-n 
4. Semimeasure: E.g., mu^M(x) = probability of guessing a halting or 
   nonhalting monotone TM program whose output starts with x 

Check out: Measures and Probability Distributions
Section 4 of "Algorithmic TOEs"

Juergen Schmidhuber

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