ISSN:
1420-8903
Keywords:
Primary 39B40
;
Secondary 94A17, 28A60
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Summary Let ℬ be a ring of sets, and letI be thek-dimensional open unit interval. The functional equation $$\begin{gathered} \varphi (E \cup F,G;p) + \mu (1 - p)\varphi \left( {E,F;\frac{q}{{1 - p}}} \right) \hfill \\ = \varphi (E \cup G,F;q) + \mu (1 - q)\varphi \left( {E,G;\frac{p}{{1 - q}}} \right), \hfill \\ \end{gathered} $$ for all disjoint triplesE, F, G of nonvoid sets in ℬ and all pairsp, q inI withp + q ∈ I, is solved for ϕ and multiplicative μ. This problem was posed by Aczél in Aequationes Math.26 (1984), 255–260. Our solution to this problem leads to an axiomatic characterization of measures of inset informationI n (E 1,⋯,E n ; $$\bar p$$ 1,⋯, $$\bar p$$ n) which have the representations $$\begin{gathered} f(E_1 ) + \sum\limits_2^n {g(E_i ) - f\left( {\bigcup\limits_1^n {E_i } } \right) + l(\bar p_1 ) + \sum\limits_1^n {\lambda (E_i ,\bar p_i )} } (for \mu = 1) \hfill \\ f(E_1 )\mu (\bar p_1 ) + \sum\limits_2^n {g(E_i )\mu (\bar p_i ) - f\left( {\bigcup\limits_1^n {E_i } } \right) + \sum\limits_1^n {l(\bar p_1 )\mu (\bar p_i )} } (for \mu additive) \hfill \\ f( \hfill \\ \end{gathered} $$ Herel is logarithmic, λ is additive in the events and logarithmic in the probabilities, andf andg are arbitrary functions. A key step in the process is to solve the equation $$J(E \cup F,G) + J(E,F) = J(E \cup G,F) + J(E,G)$$ for disjoint triplesE, F, G of nonempty sets in ℬ. A new construction was developed to handle the case where ℬ happens to be an algebra.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01836096
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