Kullback and Leibler's (1951)  measure of information associated with the probability distributions and is given by
The measure (2.1) has many names given by different authors such as, relative information, directed divergence, cross entropy, function of discrimination etc.. Here we shall refer it "relative information". It has found many applications in setting important theorems in information theory and statistics.
The Kerridge's (1961)  measure of information generally referred as inaccuracy associated with two probability distributions is given by
Various authors studied characterizations and properties of the measures (2.1) and (2.2) separately. Here we present their joint study.
Let us consider a measure
Then for , , we get (2.1) and for , , we get (2.2).
For simplicity, let define
Theorem 2.1. Let
(reals) be a function satisfying the following axioms:
Then is given by (2.3).
By considering and in (2.3) we get (2.1). Again taking and we get (2.2).
Measure (2.3) can also be characterized by different approaches using functional equation or axiomatic aprroach. In functional's equation approach, the following two equations are frequently used:
For more details refer to Mathai and Rathai (1975) ,
Autar (1975) , Taneja (1979)