×

Measures that are translation invariant in one coordinate. (English) Zbl 0617.28002

Stochastic processes, 6th Semin., Charlottesville/Va. 1986, Prog. Probab. Stat. 13, 31-34 (1987).
[For the entire collection see Zbl 0607.00014.]
Let \(\lambda\) denote a measure on (\({\mathbb{R}}\times E\), \({\mathcal B}\otimes {\mathcal E})\) (\({\mathcal B}\) Borel subsets of \({\mathbb{R}}\), (E,\({\mathcal E})\) arbitrary measurable space), which is a countable sum of finite measures and which is translation invariant in its first coordinate, i.e. \(\lambda (F_ t)=\lambda (F)\) holds for all \(t\in {\mathbb{R}}\) and any positive, measurable function on \({\mathbb{R}}\times E\), where \(F_ t\) is defined by \(F_ t(s,x)=F(s+t,x)\quad (s,t\in {\mathbb{R}},\quad x\in E).\) The author proves the following Fubini-type theorem: \[ \lambda (F)=\int_{{\mathbb{R}}}dt\int_{E}F(t,x)\mu (dx)=\int_{E}\mu (dx)\int_{{\mathbb{R}}}F(t,x)dt, \] where \(\mu\) is a measure on (E,\({\mathcal E})\), which is uniquely determined by \(\lambda (\phi \otimes f)=\mu (f)\ell (\phi)\) (\(\ell\) Lebesgue measure, \(F=\phi \otimes f)\).
Reviewer: D.Plachky

MSC:

28A15 Abstract differentiation theory, differentiation of set functions

Citations:

Zbl 0607.00014