Solutions to Exercises

Part of the Fields Institute Monographs book series (FIM, volume 35)


Let ν be a probability measure on \(\mathbb{R}\) such that \(\int _{\mathbb{R}}\vert t\vert ^{n}\,d\nu (t) <\infty\). For mn,
$$\displaystyle\begin{array}{rcl} \int _{\mathbb{R}}\vert t\vert ^{m}\,d\nu (t)& =& \int _{ \vert t\vert \leq 1}\vert t\vert ^{m}\,d\nu (t) +\int _{ \vert t\vert>1}\vert t\vert ^{m}\,d\nu (t) {}\\ & \leq & \int _{\vert t\vert \leq 1}1\,d\nu (t) +\int _{\vert t\vert>1}\vert t\vert ^{n}\,d\nu (t) {}\\ & \leq & \nu (\mathbb{R}) +\int _{\mathbb{R}}\vert t\vert ^{n}\,d\nu (t) {}\\ & <& \infty. {}\\ \end{array}$$


  1. 130.
    J.A. Mingo, R. Speicher, Schwinger-Dyson equations: classical and quantum. Probab. Math. Stat. 33(2), 275–285 (2013)MathSciNetzbMATHGoogle Scholar
  2. 138.
    A. Nica, D. Shlyakhtenko, R. Speicher, R-diagonal elements and freeness with amalgamation. Can. J. Math. 53(2), 355–381 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 152.
    W. Rudin, Functional Analysis International Series in Pure and Applied Mathematics, 2nd edn. (McGraw-Hill, Inc., New York, 1991)Google Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsQueen’s UniversityKingstonCanada
  2. 2.FB MathematikUniversität des SaarlandesSaarbrückenGermany

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