“Lebesgue measure” on $\mathbb {R}^{\infty }$, II

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 $m$ 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: $[P_{j},P_{k}]=[Q_{j},Q_{k}]=0$, $[P_{j},Q_{k}]=i\delta _{jk}$, $1\le j,k<+\infty$.