Geodesic
Krivine’s Function Calculus and Bochner Integration
Canadian mathematical bulletin, Tome 62 (2019) no. 3, pp. 663-669

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DOI

We prove that Krivine’s Function Calculus is compatible with integration. Let $(\unicode[STIX]{x1D6FA},\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D707})$ be a finite measure space, $X$ a Banach lattice, $\mathbf{x}\in X^{n}$, and $f:\mathbb{R}^{n}\times \unicode[STIX]{x1D6FA}\rightarrow \mathbb{R}$ a function such that $f(\cdot ,\unicode[STIX]{x1D714})$ is continuous and positively homogeneous for every $\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}$, and $f(\mathbf{s},\cdot )$ is integrable for every $\mathbf{s}\in \mathbb{R}^{n}$. Put $F(\mathbf{s})=\int f(\mathbf{s},\unicode[STIX]{x1D714})\,d\unicode[STIX]{x1D707}(\unicode[STIX]{x1D714})$ and define $F(\mathbf{x})$ and $f(\mathbf{x},\unicode[STIX]{x1D714})$ via Krivine’s Function Calculus. We prove that under certain natural assumptions $F(\mathbf{x})=\int f(\mathbf{x},\unicode[STIX]{x1D714})\,d\unicode[STIX]{x1D707}(\unicode[STIX]{x1D714})$, where the right hand side is a Bochner integral.
DOI : 10.4153/S0008439518000036
Mots-clés : Banach lattice, function calculus, Bochner integral 
Troitsky, V. G.; Türer, M. S. Krivine’s Function Calculus and Bochner Integration. Canadian mathematical bulletin, Tome 62 (2019) no. 3, pp. 663-669. doi: 10.4153/S0008439518000036
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     author = {Troitsky, V. G. and T\"urer, M. S.},
     title = {Krivine{\textquoteright}s {Function} {Calculus} and {Bochner} {Integration}},
     journal = {Canadian mathematical bulletin},
     pages = {663--669},
     year = {2019},
     volume = {62},
     number = {3},
     doi = {10.4153/S0008439518000036},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008439518000036/}
}
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