``Lebesgue measure' on <IMG BORDER="0" SRC="/proc/2004-132-09/S0002-9939-04-07372-1/gif-title0/img1.gif" >, II期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

按检索

``Lebesgue measure' on , II

Authors:	Richard L Baker

Institution:	Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242

Abstract:	Let $\mathbb{R}$ be the set of real numbers, and define $\mathbb{R}^{\infty }=\prod \limits ^{\infty }_{i=1}\mathbb{R}$ . We construct a complete measure space $(\mathbb{R}^{\infty },\mathcal{L},\lambda )$ where the $\sigma$ -algebra $\mathcal{L}$ contains the Borel subsets of $\mathbb{R}^{\infty }$ , and $\lambda$ is a translation-invariant measure such that for any measurable rectangle $R=\prod \limits ^{\infty }_{i=1}R_{i}$ , if $0\le \prod \limits ^{\infty }_{i=1}m(R_{i})<+\infty$ , then $\lambda (R)=\prod \limits ^{\infty }_{i=1}m(R_{i})$ , where is Lebesgue measure on $\mathbb{R}$ . The measure $\lambda$ is not $\sigma$ -finite. We prove three Fubini theorems, namely, the Fubini theorem, the mean Fubini-Jensen theorem, and the pointwise Fubini-Jensen theorem. Finally, as an application of the measure $\lambda$ , we construct, via selfadjoint operators on $L_{2}(\mathbb{R}^{\infty },\mathcal{L},\lambda )$ , a ``Schrödinger model' of the canonical commutation relations: $P_{j},P_{k}]=Q_{j},Q_{k}]=0$ , $P_{j},Q_{k}]=i\delta _{jk}$ , $1\le j,k<+\infty$ .

Keywords:	Canonical commutation relations Elliott-Morse measures Fubini theorem Fubini-Jensen theorem infinite-dimensional Lebesgue measure invariant measures Schr\"{o}dinger model

	点击此处可从《Proceedings of the American Mathematical Society》浏览原始摘要信息
	点击此处可从《Proceedings of the American Mathematical Society》下载免费的PDF全文

设为首页 | 免责声明 | 关于勤云 | 加入收藏