论文标题
与过滤器集成
Integration with filters
论文作者
论文摘要
我们介绍了一个有限集的家族的过滤器定义的集成概念。该过程对应于确定范围在于有限平均值的任何代数结构的函数的平均值。如此确定的平均值在于原始功能范围的适当扩展。最相关的情况涉及扩展有理数领域的代数结构。因此，可以将上部和下标准部分与滤波器积分相关联。这些数字可以解释为上限和下限，这是人们期望从经验上观察到的函数的平均值。我们讨论了滤波器积分的主要属性，并表明它具有足够的表现力来表示每个实际积分。作为应用程序，我们在无限维矢量空间上定义了几何度量，该量子量克服了一些已知的限制，该局限性在实用值衡量标准中有效。我们还讨论了如何将滤波器积分应用于非Archimedean集成问题，并为这些积分开发迭代理论。
We introduce a notion of integration defined from filters over families of finite sets. This procedure corresponds to determining the average value of functions whose range lies in any algebraic structure in which finite averages make sense. The average values so determined lie in a proper extension of the range of the original functions. The most relevant scenario involves algebraic structures that extend the field of rational numbers; hence, it is possible to associate to the filter integral an upper and lower standard part. These numbers can be interpreted as upper and lower bounds on the average value of the function that one expects to observe empirically. We discuss the main properties of the filter integral and we show that it is expressive enough to represent every real integral. As an application, we define a geometric measure on an infinite-dimensional vector space that overcomes some of the known limitations valid for real-valued measures. We also discuss how the filter integral can be applied to the problem of non-Archimedean integration, and we develop the iteration theory for these integrals.
