A general form of the covering principle and relative differentiation of additive functions. II
A general form of the covering principle and relative differentiation of additive functions. II
复制标题
加性函数的覆盖原理和相对微分的一般形式。
DOI：
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发表时间：
1946
影响因子：
0.8
通讯作者：
Thomas F. Reynolds
中科院分区：
文献类型：
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作者：
C. Begley;C. H. Slater;M. Engel;Thomas F. Reynolds
在我的论文下相同的标题，我将提到在未来作为我的，我推广了维塔利覆盖原则的情况下勒贝格措施的情况下，任何非负的加法函数。这使我能够建立加法函数的相对微分。在覆盖原理的这种广义形式中，集合的收敛序列被限制为同心圆序列，因此所得到的微分是对称意义上的微分。本文将这一原理推广到覆盖集的任何正则收敛序列的情形，从而建立了一般意义上的可加函数的相对微分，特别是不定积分关于任何测度函数的微分。这个问题有一个完整的解决方案。不定积分几乎在所有点上都是可微的。在一般测度函数的情况下，导数在几乎所有点上都等于被积函数是不正确的，但给出了使其为真的充分必要条件。
In my paper under the same title, to which I shall refer in future as I‡, I generalized the Vitali covering principle from the case of Lebesgue measure to the case of any non-negative additive function. This allowed me to establish the relative differentiation of additive functions. The convergent sequences of sets in this generalized form of the covering principle were restricted to sequences of concentric circles, and therefore the differentiation arrived at was that in the symmetrical sense. In the present paper, I extend the principle to the case of any regular convergent sequences of covering sets; and then establish the relative differentiation of additive functions in the general sense, and in particular the differentiation of indefinite integrals with respect to any measure function. This problem has a complete solution. It is established that indefinite integrals are differentiable at almost all points. In the case of the general measure function, it is not true that the derivative is equal to the integrand at almost all points, but necessary and sufficient conditions are given under which this is true.