"Lebesgue Measure" on R ∞

Richard Baker
1991 Proceedings of the American Mathematical Society  
Let R be the set of real numbers, and define R ∞ = ∞ i=1 R. We construct a complete measure space (R ∞ , L, λ) where the σ-algebra L contains the Borel subsets of R ∞ , and λ is a translation-invariant measure such that where m is Lebesgue measure on R. The measure λ is not σfinite. 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 λ, we construct, via selfadjoint
more » ... selfadjoint operators on L 2 (R ∞ , L, λ), a "Schrödinger model" of the canonical commutation relations: [P j , P k ] = [Q j , Q k ] = 0, [P j , Q k ] = iδ jk , 1 ≤ j, k < +∞.
doi:10.2307/2048779 fatcat:d4lodmla3fgh5er74cvlg4aana