ISSN:
1618-1891
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Summary By means of basic properties of the functions min{x, y} and max{x, y} andsum supremum and sum infimum functionals, an elementary argument for a nonintegrable form of the Lipschitz portion of the Bochner-Radon-Nikodym Theorem is obtained. It is also shown that if a 〈 b and h is a function from [a; b] into ℝ,then the following two statements are equivalent: 1) there is a set U, a field F of subsets of U, a function α from F into exp ([a; b]) and a real nonnegative-valued finitely additive function μ on F such that if a⩽p 〈q ⩽ b, then $$\int\limits_U {\max } \left\{ {\min \left\{ {\alpha (I) - p,q - p} \right\},0} \right\}\mu (I)$$ exist and is h(q) − h(p), and 2) for some M 〉 0, if a〈r〈s〈t〈b, then M〉 (h(s)−h(r))/(s−r)〉 (h(t)−h(s))/(t−)〉0
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01765935
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