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Vector gauging

Vector gauging

The vector gauging (vector そくど British: vector measure) in the field of the mathematics is the vector function with a particular property defined on a certain interrelated groups. It is the generalization of the concept of the gauging to take only a non-minus number actual number.

Table of contents

A definition and the first conclusion

Aggregate  バナッハ space   But, when was given; with the limited additive vector gauging (or briefly gauging),  Bare な gathers in each other of the inner の option    It is

 

But, the function that is managed   I say a saw.

Vector gauging   But, to be like countable addition,  It is the line of the bare な meeting for each other of the inner の option   でその merger   For the thing which に is included in,

 

But, I say that it is managed. But the series of the right side is バナッハ space   I shall converge about の norm.

Additive vector gauging   But, the necessary and sufficient condition to be like countable addition is any line such as the statement above   It is

 

But, it is to be managed. Here    It is の norm.

The countable additive vector gauging defined on σ-algebra is more common than gauging and mark gauging belonging to, double bare gauging. But they are extended sections each  It is a countable additive function to take a value on the set of, real number and the set of the complex number.

Example

Section   And family of all ルベーグ measurability meetings included in the section   I think about the aggregate which から becomes. Any such meeting   It is

 

I define を. Here    It is の instructions function. この   But, two following different results produce it which space you take a value in.

  •   から Lp space   When I was considered to be への function,  It is the vector gauging that is not は 可算加法的.
  •   から Lp space   When I was considered to be への function,  It is the vector gauging that is は 可算加法的.

These statements easily obey it from above-mentioned condition (*).

Variation of the vector gauging

Vector gauging   に vs. do; the variation (variation) 

 

It is defined によって. Here the upper limit of the right side,  Inner のすべての   It is all division to bare な meeting for each other of the に vs. してその limited number

 

取 られる where it is. Again    It is an upper norm.  The の variation   It is a limited additive function to take a に level.  Of the inner の option   It is

 

But, it is established.  But, if is limited; gauging   It is said that I belong to は 有界変分 (bounded variation).  But, if it is the vector gauging of the existence world variation,  But, with the thing that is like countable addition   But, it is the equivalent to be like countable addition.

Theorem of リャプノフ

According to the theorem of リャプノフ in the theory of the vector gauging, the range of the vector gauging is shut and a convex (is non-atomic (English version)) [1]; [2] [3]. Actually, a range of the non-atomic vector gauging is ゾノイド (extreme shut convex set of the convergence line of ゾノトープ); [2]. It is used mathematical economics [4] [5] [6] and a Big Bang control theory [1] for this theorem in [8] and statistics theory (English version) [8] [3] [7]. The theorem of リャプノフ is proved by using the lemma (English version) of the Shah play = fork man considered to be the disintegration resemblance [9]; [8] [10] [11]

Footnote

  1. ^ a b Kluvánek, I., Knowles, G., Vector Measures and Control Systems, North-Holland Mathematics Studies 20, Amsterdam, 1976.
  2. ^ a b Diestel, Joe; Uhl, (1977). Vector measures. Providence, R.I: American Mathematical Society. ISBN 0-8218-1515-6. 
  3. ^ a b Rolewicz, Stefan (1987). Functional analysis and control theory: Linear systems. Mathematics and its Applications (East European Series). 29 (Translated from the Polish by Ewa Bednarczuk ed.). Dordrecht; Warsaw: D. Reidel Publishing Co. PWN-Polish Scientific Publishers. pp. xvi+524. ISBN 90-277-2186-6. MR 920371. OCLC 13064804 
  4. ^ Roberts, John (July 1986). "Large economies". In David M. Kreps. Contributions to the New Palgrave. Research paper. 892. Palo Alto, CA: Graduate School of Business, Stanford University. pp. 30–35. (Draft of articles for the first edition of New Palgrave Dictionary of Economics). https://gsbapps.stanford.edu/researchpapers/library/RP892.pdf 7 February 2011 reading. 
  5. ^ Aumann, Robert J. (January 1966). "Existence of competitive equilibrium in markets with a continuum of traders." Econometrica 34 (1): 1–17. JSTOR 1909854. MR 191623.  This paper builds on two papers by Aumann: "Markets with a continuum of traders." Econometrica 32 (1–2): 39–50. (January–April 1964). JSTOR 1913732. MR 172689.  "Integrals of set-valued functions." Journal of Mathematical Analysis and Applications 12 (1): 1–12. (August 1965). doi: 10.1016/0022-247X(65) 90,049-1. MR 185073. 
  6. ^ Vind, Karl (May, 1964). "Edgeworth-allocations in an exchange economy with many traders." International Economic Review 5 (2): pp. 165–77 Vind's article was noted by Debreu (1991, p. 4) with this comment:

    The concept of a convex set (i.e., a set containing the segment connecting any two of its points) had repeatedly been placed at the center of economic theory before 1964. It appeared in a new light with the introduction of integration theory in the study of economic competition: If one associates with every agent of an economy an arbitrary set in the commodity space and if one averages those individual sets over a collection of insignificant agents, then the resulting set is necessarily convex. [Debreu appends this footnote: "On this direct consequence of a theorem of A. A. Lyapunov, see Vind (1964)."] But explanations of the ... functions of prices ... can be made to rest on the convexity of sets derived by that averaging process. Convexity in the commodity space obtained by aggregation over a collection of insignificant agents is an insight that economic theory owes ... to integration theory. [Italics added]

    Debreu, Gérard (March, 1991). "The Mathematization of economic theory." The American Economic Review 81 (Presidential address delivered at the 103rd meeting of the American Economic Association, 29 December 1990, Washington, DC) 

  7. ^ Hermes, Henry; LaSalle, Joseph P. (1969). Functional analysis and time optimal control. Mathematics in Science and Engineering. 56. New York-London: Academic Press. pp. viii+136. MR 420366. 
  8. ^ a b c Artstein, Zvi (1980). "Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points." SIAM Review 22 (2): pp. 172–185. doi: 10.1137/1022026 
  9. ^ Tardella, Fabio (1990). "A new proof of the Lyapunov convexity theorem." SIAM Journal on Control and Optimization 28 (2): pp. 478–481. doi: 10.1137/0328026 
  10. ^ Starr, Ross M. (2008). "Shapley–Folkman theorem". In Durlauf, Steven N.; Blume, Lawrence E., ed.. The New Palgrave Dictionary of Economics (Second ed.). Palgrave Macmillan. pp. 317–318 (1st ed.). doi: 10.1057/9780230226203.1518. http://www.dictionaryofeconomics.com/article?id=pde2008_S000107. 
  11. ^ Page 210 Mas-Colell, Andreu (1978). "A note on the core equivalence theorem: How many blocking coalitions are there?." Journal of Mathematical Economics 5 (3): pp. 207–215. doi: 10.1016/0304-4068(78) 90,010-1 

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