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	In ~~[[mathematics]]~~, ~~particularly in~~ the ~~area of [[functional analysis]]~~ and ~~[[topological vector space]]~~s, ~~the '''vague topology''' is~~ an ~~example~~ of ~~the [[weak topology\|weak~~-* topology]] which arises in ~~the study~~ of ~~[[measure theory\|measures]]~~ on ~~[[locally compact Hausdorff space]]s.~~		In reality, many small business owners don"t understand the method or complain that doing an industry research is too complex or too expensive and wonder why or if it is necessary. <br><br>What"s market research? <br><br>In one of the most basic terms, market analysis can be an assessment of: <br><br>- A certain problem or opportunity in an industry. <br><br>- The n... <br><br>The definition of "market analysis" is usually puzzling to entrepreneurs, particularly for individuals who focus on a particular market or market segment. <br><br>Actually, many small businesses do not comprehend the process or protest that conducting a market analysis is too complicated or too expensive and wonder why or if it"s necessary. <br><br>What"s market research? <br><br>In one of the most basic terms, a market research is an examination of: <br><br>- A particular problem or opportunity in market. 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	~~Let ''X'' be~~ a ~~[[locally compact Hausdorff space]]~~. ~~Let ''M''(''X'') be~~ the ~~space of [[complex numbers\|complex]] [[Radon measure]]~~s ~~on ''X'', and ''C''~~<~~sub~~>0<~~/sub~~>~~(''X'')~~<~~sup~~><~~/sup~~> ~~denote~~ the ~~dual~~ of ~~''C''~~<~~sub~~>0<~~/sub~~>~~(''X''), the~~ [~~[Banach space]~~] of ~~complex [[continuous function]]s on ''X'' [[vanish at infinity\|vanishing at infinity]] equipped with~~ the ~~[[uniform norm]]~~. By the ~~[[Riesz representation theorem]] ''M''(''X'') is [[isometry\|isometric]] to ''C''~~<~~sub~~>0<~~/sub~~>~~(''X'')~~<~~sup~~><~~/sup~~>. ~~The isometry maps~~ a ~~measure ''~~μ~~'' to~~ a [~~[linear functional~~]]

	:<~~math~~>~~I_\mu(f) := \int_X f\~~, ~~d\mu~~.<~~/math~~>

	~~The '''vague topology'''~~ is the ~~[[Weak topology\|weak-* topology]] on ''C''~~<~~sub~~>0<~~/sub~~>~~(''X'')~~<~~sup~~><~~/sup~~>. ~~The corresponding topology on ''M''~~(~~''X''~~) ~~induced~~ by the ~~isometry~~ from ~~''C''~~<~~sub~~>0<~~/sub~~>~~(''X'')~~<~~sup~~><~~/sup~~> is ~~also called~~ the ~~vague topology on ''M''(''X'')~~. ~~Thus,~~ in ~~particular, one may refer~~ to ~~vague convergence~~ of ~~measure ''μ''~~<~~sub~~>~~''n''~~<~~/sub~~> &~~rarr~~; ''μ~~''.~~

	~~One application of this is to [[probability theory]]: for example, the [[central limit theorem]] is essentially a statement that if ''~~μ''<~~sub~~>~~''n''~~<~~/sub~~> are ~~the [[probability measure]]s~~ for ~~certain sums of [[independent random variables]], then ''μ''~~<~~sub~~>~~''n''~~<~~/sub~~> ~~converge weakly to a [[normal distribution]], i.e~~. ~~the measure ''~~μ''<~~sub~~>~~''n''~~<~~/sub~~> is ~~"approximately normal"~~ for ~~large ''n''.~~

	~~==References==~~
	* {{citation\|author=Dieudonné, ~~Jean\|authorlink=Jean Dieudonné\|chapter=§13.4. The vague topology\|title=Treatise on analysis\|volume=II\|publisher=Academic Press\|year=1970}}.~~
	* G.B. Folland, ~~Real Analysis: Modern Techniques~~ and ~~Their Applications, 2nd ed, John Wiley & Sons, Inc~~.~~, 1999.~~

	~~{{PlanetMath attribution\|id=7212\|title=Weak-* topology of the space of Radon measures}}~~

	[~~[Category~~:~~Real analysis]]~~
	~~[[Category:Topology of function spaces]~~]

en>CBM: Removing stub tag(s)

2010-02-21T21:20:02Z

Removing stub tag(s)

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In [[mathematics]], particularly in the area of [[functional analysis]] and [[topological vector space]]s, the '''vague topology''' is an example of the [[weak topology|weak-* topology]] which arises in the study of [[measure theory|measures]] on [[locally compact Hausdorff space]]s.

Let ''X'' be a [[locally compact Hausdorff space]]. Let ''M''(''X'') be the space of [[complex numbers|complex]] [[Radon measure]]s on ''X'', and ''C''0(''X'')* denote the dual of ''C''0(''X''), the [[Banach space]] of complex [[continuous function]]s on ''X'' [[vanish at infinity|vanishing at infinity]] equipped with the [[uniform norm]]. By the [[Riesz representation theorem]] ''M''(''X'') is [[isometry|isometric]] to ''C''0(''X'')*. The isometry maps a measure ''μ'' to a [[linear functional]]

:<math>I_\mu(f) := \int_X f\, d\mu.</math>

The '''vague topology''' is the [[Weak topology|weak-* topology]] on ''C''0(''X'')*. The corresponding topology on ''M''(''X'') induced by the isometry from ''C''0(''X'')* is also called the vague topology on ''M''(''X''). Thus, in particular, one may refer to vague convergence of measure ''μ''''n'' → ''μ''.

One application of this is to [[probability theory]]: for example, the [[central limit theorem]] is essentially a statement that if ''μ''''n'' are the [[probability measure]]s for certain sums of [[independent random variables]], then ''μ''''n'' converge weakly to a [[normal distribution]], i.e. the measure ''μ''''n'' is "approximately normal" for large ''n''.

==References==
* {{citation|author=Dieudonné, Jean|authorlink=Jean Dieudonné|chapter=§13.4. The vague topology|title=Treatise on analysis|volume=II|publisher=Academic Press|year=1970}}.
* G.B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999.

{{PlanetMath attribution|id=7212|title=Weak-* topology of the space of Radon measures}}

[[Category:Real analysis]]
[[Category:Topology of function spaces]]

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72.226.86.106: there's a page for this

en>CBM: Removing stub tag(s)