Sublinear functions of measures and variational integrals
Sublinear functions of measures and variational integrals
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测度和变分积分的次线性函数
DOI：
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发表时间：
1964
期刊：
影响因子：
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通讯作者：
J. Serrin
中科院分区：
文献类型：
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作者：
C. Goffman;J. Serrin
本文的最初目的是建立某些非参数变分积分可以被认为是可容许函数定义域上的测度。这是通过展示一个明确的公式所涉及的功能，该公式包含本身，在案件的表面积积分，证明都Tonelli的著名定理勒贝格面积和一个不太知名的，但更深层次的结果Verchenko。它很快就变得明显，然而，基本技术是相当普遍的，并可以，事实上，被措辞完全脱离变分微积分只是作为一种方法产生措施方面的其他措施。我们将遵循这一更抽象的过程中的文件，只有直接转向变分在最后一节。考虑基集S的子集的g-环R上的可数可加集函数。假设函数g在真实的Banach空间中有其值，对于特定的应用，它可以是真实的数R1，或欧几里得数空间R。设（p）是上的有界次线性泛函.这意味着，对于所有p，q，
The original purpose of this paper was to establish that certain non-parametric variational integrals may be considered as measures on the domain of definition of the admissible functions. This was accomplished by exhibiting an explicit formula for the functional involved, the formula containing within itself, in the case of the surface area integral, a proof both of Tonelli’s celebrated theorem on Lebesgue area and of a less well known but deeper result of Verchenko. It soon became apparent, however, that the basic techniques were considerably more general, and could, in fact, be phrased entirely apart from variational calculus simply as a method of generating measures in terms of other measures. We shall follow this more abstract course in the paper, and only turn directly to variational calculus in the final section. Consider a countably additive set function on a g-ring R of subsets of a basic set S. The function g will be supposed to have its values in a real Banach space , which for particular applications may be the real numbers R1, or the Euclidean number space R. Let (p) be a bounded sublinear functional on into the reals. By this we mean that for all p, q in