Abstract
The collection of ordered n-tuples of real numbers (x1,...,xn) is denoted by Rn. We sometimes call Rn n-dimensional space; in doing so, we will refer to the n-tuples as the coordinates of a point. If n = 1, R 1 is the set of real numbers.
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If M 1⫅M 2, but M 1≠M 2, then we write M 1⊂M 2.
Whenever the equality sign occurs in the definition of these intervals it is supposed that-∞ <a respectively b<∞.
This property characterizes the compact sets and is frequently used as definition of compactness which, in this form, can easily be carried over to more general spaces.
cM is also called the characteristic function of M.
For more detailed information, see P. Halmos, Measure Theory, D. Van Nostrand, New York 1950, and H. Richter, Wahrscheinlichkeitstheorie, Second Edition, Bd. 66 Springer-Verlag, Berlin-New York 1966.
Cf. A. Ljapunov, Bull. Acad. Sci. URSS, Ser. Math. 4,465–478 (1940). See also J. Lindenstrauss, J. Math. Mech. 15, 971–972 (1966).
This theorem may be found, for example, in the book by K. Krickeberg, Probability Theory, Addison-Wesley, Reading-London, 1965.
This inequality sometimes goes under the name of Bunjakowski.
More generally, we have the Hölder inequality: \( \int\limits_R {|fg|d\mu \le \left( {\int\limits_R {|f|{\,^p}d\mu } } \right){{\left( {\int\limits_R {|g|{\,^q}d\mu } } \right)}^{1/q}}} \) with p,q>1 and 1/p + 1/q =1.
This was first carried out by J. Radon: J. Radon, Österreich. Akad. Wiss., math.-naturw. Kl., S.-Ber. 122, Abt. IIa, 1295–1438 (1913).
For example, see E. L. Lehmann, Testing Statistical Hypotheses, John Wiley & Sons, New York 1959, 354. See also G. Nolle und D. Plachky, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 8, 182–184 (1967).
The proof takes the following form: First we notice immediately that \( c\int\limits_M {k(x)dx\, = \,\int\limits_M {ck(x)dx} } \), for any complex constant c. then, setting \( \phi = \arctan \left( {\left( {\int\limits_M {k(x)dx} } \right)/\Re \left( {\int\limits_M {k(x)dx} } \right)} \right) \), we have \( |\int\limits_M {k(x)dx} |\, = \Re \left( {{e^{ - i\phi }}\int\limits_M {k(x)dx} } \right)\, = \,\int\limits_M \Re ({e^{ - i\phi }}k(x))dx \le \int\limits_M {|k(x)|dx} \).
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© 1974 Springer-Verlag Berlin · Heidelberg
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Schmetterer, L. (1974). Notation and Preliminary Remarks. In: Introduction to Mathematical Statistics. Die Grundlehren der mathematischen Wissenschaften, vol 202. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-65542-5_1
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DOI: https://doi.org/10.1007/978-3-642-65542-5_1
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