Notation and Preliminary Remarks

Chapter
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 202)

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.

Keywords

Measurable Space Pairwise Disjoint Inverse Image Arbitrary Real Number Preliminary Remark
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References

1. 1.
If M 1M 2, but M 1M 2, then we write M 1M 2.Google Scholar
2. 2.
Whenever the equality sign occurs in the definition of these intervals it is supposed that-∞ <a respectively b<∞.Google Scholar
3. 3.
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.Google Scholar
4. 4.
cM is also called the characteristic function of M.Google Scholar
5. 5.
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.Google Scholar
6. 6.
7. 7.
This theorem may be found, for example, in the book by K. Krickeberg, Probability Theory, Addison-Wesley, Reading-London, 1965.Google Scholar
8. 8.
This inequality sometimes goes under the name of Bunjakowski.Google Scholar
9. 9.
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.Google Scholar
10. 10.
This was first carried out by J. Radon: J. Radon, Österreich. Akad. Wiss., math.-naturw. Kl., S.-Ber. 122, Abt. IIa, 1295–1438 (1913).
11. 11.
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).Google Scholar
12. 12.
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 $$|\int\limits_M {k(x)dx} |\, = \Re \left( {{e^{ - i\phi }}\int\limits_M {k(x)dx} } \right)\, = \,\int\limits_M \Re ({e^{ - i\phi }}k(x))dx \le \int\limits_M {|k(x)|dx}$$. Google Scholar