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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Daniell integral ： ウィキペディア英語版 Daniell integral In mathematics, the Daniell integral is a type of integration that generalizes the concept of more elementary versions such as the Riemann integral to which students are typically first introduced. One of the main difficulties with the traditional formulation of the Lebesgue integral is that it requires the initial development of a workable measure theory before any useful results for the integral can be obtained. However, an alternative approach is available, developed by that does not suffer from this deficiency, and has a few significant advantages over the traditional formulation, especially as the integral is generalized into higher-dimensional spaces and further generalizations such as the Stieltjes integral. The basic idea involves the axiomatization of the integral. ==Axioms== We start by choosing a family $H$ of bounded real functions (called ''elementary functions'') defined over some set $X$, that satisfies these two axioms: * $H$ is a linear space with the usual operations of addition and scalar multiplication. * If a function $h(x)$ is in $H$, so is its absolute value $|h(x)|$. In addition, every function ''h'' in ''H'' is assigned a real number $Ih$, which is called the ''elementary integral'' of ''h'', satisfying these three axioms: * Linearity : If ''h'' and ''k'' are both in H, and $\backslash alpha$ and $\backslash beta$ are any two real numbers, then $I(\backslash alpha\; h\; +\; \backslash beta\; k)\; =\; \backslash alpha\; Ih\; +\; \backslash beta\; Ik$. * Nonnegativity : If $h(x)\; \backslash ge\; 0$, then $Ih\; \backslash ge\; 0$. * Continuity : If $h\_n(x)$ is a nonincreasing sequence (i.e. $h\_1\; \backslash ge\; \backslash cdots\; \backslash ge\; h\_k\; \backslash ge\; \backslash cdots$) of functions in $H$ that converges to 0 for all $x$ in $X$, then $Ih\_n\; \backslash to\; 0$. That is, we define a continuous non-negative linear functional $I$ over the space of elementary functions. These elementary functions and their elementary integrals may be any set of functions and definitions of integrals over these functions which satisfy these axioms. The family of all step functions evidently satisfies the above axioms for elementary functions. Defining the elementary integral of the family of step functions as the (signed) area underneath a step function evidently satisfies the given axioms for an elementary integral. Applying the construction of the Daniell integral described further below using step functions as elementary functions produces a definition of an integral equivalent to the Lebesgue integral. Using the family of all continuous functions as the elementary functions and the traditional Riemann integral as the elementary integral is also possible, however, this will yield an integral that is also equivalent to Lebesgue's definition. Doing the same, but using the Riemann–Stieltjes integral, along with an appropriate function of bounded variation, gives a definition of integral equivalent to the Lebesgue–Stieltjes integral. Sets of measure zero may be defined in terms of elementary functions as follows. A set $Z$ which is a subset of $X$ is a set of measure zero if for any $\backslash epsilon\; >\; 0$, there exists a nondecreasing sequence of nonnegative elementary functions $h\_p(x)$ in ''H'' such that and $$ \sup_p h_p(x) \ge 1 on $Z$. A set is called a set of full measure if its complement, relative to $X$, is a set of measure zero. We say that if some property holds at every point of a set of full measure (or equivalently everywhere except on a set of measure zero), it holds almost everywhere. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Daniell integral」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.