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Hartley Function
Definition
The Hartley function is a measure of uncertainty, introduced by Hartley in 1928. If we pick a sample from a finite set
uniformly at random, the information revealed after we know the outcome is given by the Hartley function
Remark:
The Hartley function is a special case of Shannon's entropy. Each element in the sample space 
is associated with probability 
. For an element
, the Hartley information of the event
is
, which is constant over
. The average
information over the whole sample space is thus also equal to 
.
The Hartley function only depends on the number of elements in a set, and hence can be viewed as a function on natural numbers. Rényi showed that the Hartley function in base 2 is the only function mapping natural numbers to real numbers that satisfies
-
(additivity),
-
(monotonicity), and
(normalization).
Condition 1 says that the uncertainty of the Cartesian product
of
two finite sets 
and 
is the sum
of uncertainties of 
and
. Condition 2 says that a larger set has larger uncertainty.