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

Latest revision as of 16:54, 28 October 2014

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.

What"s market research?

In one of the most basic terms, market analysis can be an assessment of:

- A certain problem or opportunity in an industry.

- The n...

The definition of "market analysis" is usually puzzling to entrepreneurs, particularly for individuals who focus on a particular market or market segment.

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.

What"s market research?

In one of the most basic terms, a market research is an examination of:

- A particular problem or opportunity in market. Identify additional resources on a related web resource - Click here: analysis.

- The needs of the prospective market relating to the problem or opportunity.

- Some ideas for marketing a certain products or services that fills the requirements of the target market.

When should you perform an industry analysis?

When you"re starting a small business -.

- When you are entering a new industry.

- When you"re considering a fresh service or product.

Thinking about conduct market research?

- To reduce business risks.

- To know the opportunities and problems.

- To identify sales opportunities. This unique guide to fundable competition site has some astonishing tips for when to see this belief.

- To plan your marketing/sales strategy.

The process of conducting a market research could be split into

three parts:

Part 1 - Knowledge Market Conditions

This gives you essential information regarding your entire market -- the size, your competition, the customers.

Part 2 - Determining Market Possibilities

This gives you more focused information about potential problems or opportunities in the potential market, and contains information about external factors, current and future developments, growth and more information about opponents.

Part 3 - Developing Market-Driven Methods

Here"s where we enter into what general market trends does for-you. It will help you to determine opportunities to develop your company. By understanding the industry and knowing what opportunities are available it is possible to develop a marketing strategy that leaves your competition in the dust!

Here are 10 questions that will help you obtain started:

1. What is industry I do want to reach?

- Who are they? (Basic Demographics)

- What is their biggest problem in terms of the forex market?

- Are their requirements being met by these products or services provided in forex trading?

2. Who"s my opposition in the forex market?

- Are they effective in this market?

- Are they promoting a similar product or service?

- What is the market share of the three biggest competitors in this market?

3. Can there be room for development in the forex market?

4. What is how big forex trading?

- Will there be room for development?

- Could be the industry growing? Firm? Saturated? Risky? Decreasing?

5. How is my product or service different from your competition?

6. How to reach this market?

- How is my opposition currently reaching forex trading?

- Can it be the most effective way?

- What are the alternative means of reaching the forex market?

7. What are the business models of my opposition in the forex market?

- Are they effective?

- Will there be ways to do it differently or better?

8. What do customers expect from this form of product or service?

- What are the core skills with this service or product?

- What could make the item "new" "unique" or "better" for your customer?

9. Just how much are consumers willing to pay for this product or service?

10. What"s our competitive advantage in forex trading?

Knowing the answers to these questions will not only help you figure out if there is a requirement for your product or service, it will help you figure out the best methods to price your products and services or service, reach your visitors and ultimately make more sales!.

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