Analemmatic sundial

In mathematics, a content is a real function $\mu$ defined on a field of sets ${\mathcal {A}}$ such that

1. $\mu (A)\in \ [0,\infty ]{\mbox{ whenever }}A\in {\mathcal {A}}.$
2. $\mu (\varnothing )=0.$
3. $\mu (A_{1}\cup A_{2})=\mu (A_{1})+\mu (A_{2}){\mbox{ whenever }}A_{1},A_{2}\in {\mathcal {A}}{\mbox{ and }}A_{1}\cap A_{2}=\varnothing .$

An example of a content is a measure, which is a σ-additive content defined on a σ-field. Every (real-valued) measure is a content, but not vice-versa. Contents give a good notion of integrating bounded functions on a space but can behave badly when integrating unbounded functions, while measures give a good notion of integrating unbounded functions.

Examples

An example of a content that is not a measure on a σ-algebra is the content on all subset of the positive integers that has value 1/n on the integer n and is infinite on any infinite subset.

An example of a content on the positive integers that is always finite but is not a measure can be given as follows. Take a positive linear functional on the bounded sequences that is 0 if the sequence has only a finite number of nonzero elements and takes value 1 on the sequence 1, 1, 1, ...., so the functional in some sense gives an "average value" of any bounded sequence. (Such a functional cannot be constructed explicitly, but exists by the Hahn-Banach theorem.) Then the content of a set of positive integers is the average value of the sequence that is 1 on this set and 0 elsewhere. Informally, one can think of the content of a subset of integers as the "chance" that a randomly chosen integer lies in this subset (though this is not compatible with the usual definitions of chance in probability theory, which assume countable additivity).

Integration of bounded functions

In general integration of functions with respect to a content does not behave well. However there is a well-behaved notion of integration provided that the function is bounded and the total content of the space is finite, given as follows.

Suppose that the total content of a space is finite. If f is a bounded function on the space such that the inverse image of any open subset of the reals has a content, then we can define the integral of f with respect to the content as

$\int fd\lambda =\lim \sum _{i=1}^{n}f(\alpha _{i})\lambda (f^{-1}(A_{i}))$

where the A_i form a finite collections of disjoint half-open sets whose union covers the range of f, and α_i is any element of A_i, and where the limit is taken as the diameters of the sets A_i tend to 0.

Duals of spaces of bounded functions

Suppose that μ is a measure on some space X. The bounded measurable functions on X form a Banach space with respect to the supremum norm. The positive elements of the dual of this space correspond to bounded contents λ ον Χ, with the value of λ on f given by the integral ∫fdλ. Similarly one can form the space of essentially bounded functions, with the norm given by the essential supremum, and the positive elements of the dual of this space are given by bounded contents that vanish on sets of measure 0.

Construction of a measure from a content

There are several ways to construct a measure μ from a content λ on a topological space. This section gives one such method for locally compact Hausdorff spaces such that the content is defined on all compact subsets. In general the measure is not an extension of the content, as the content may fail to be countably additive, and the measure may even be identically zero even if the content is not.

First restrict the content to compact sets. This gives a function λ of compact sets C with the following properties:

1. $\lambda (C)\in \ [0,\infty ]$ for all compact sets C
2. $\lambda (\varnothing )=0.$
3. $\lambda (C_{1})\leq \lambda (C_{2}){\mbox{ whenever }}C_{1}\subset C_{2}$
4. $\lambda (C_{1}\cup C_{2})\leq \lambda (C_{1})+\lambda (C_{2})$ for all pairs of compact sets
5. $\lambda (C_{1}\cup C_{2})=\lambda (C_{1})+\lambda (C_{2})$ for all pairs of disjoint compact sets.

There are also examples of functions λ as above not constructed from contents. An example is given by the construction of Haar measure on a locally compact group. One method of constructing such a Haar measure is to produce a left-invariant function λ as above on the compact subsets of the group, which can then be extended to a left-invariant measure.

Definition on open sets

Given λ as above, we define a function μ on all open sets by

$\mu (U)=\sup _{C\subset U}\lambda (C)$.

This has the following properties:

1. $\mu (U)\in \ [0,\infty ]$
2. $\mu (\varnothing )=0.$
3. $\mu (U_{1})\leq \mu (U_{2}){\mbox{ whenever }}U_{1}\subset U_{2}$
4. $\mu (\bigcup _{n}U_{n})\leq \oplus _{n}\lambda (U_{n})$ for any collection of open sets.
5. $\mu (\bigcup _{n}U_{n})=\oplus _{n}\lambda (U_{n})$ for any collection of disjoint open sets

Definition on all sets

Given μ as above, we extend the function μ to all subsets of the topological space by

$\mu (A)=\inf _{A\subset U}\mu (U)$.

This is an outer measure, in other words it has the following properties:

1. $\mu (A)\in \ [0,\infty ]$
2. $\mu (\varnothing )=0.$
3. $\mu (A_{1})\leq \mu (A_{2}){\mbox{ whenever }}A_{1}\subset A_{2}$
4. $\mu (\bigcup _{n}A_{n})\leq \oplus _{n}\lambda (A_{n})$ for any countable collection of sets.

Construction of a measure

The function μ above is an outer measure on the family of all subsets. Therefore it becomes a measure when restricted to the measurable subsets for the outer measure, which are the subsets E such that μ(X) = μ(X∩E) + μ(X\E) for all subsets X.. If the space is locally compact then every open set is measurable for this measure.

The measure μ does not necessarily coincide with the content λ on compact sets, However it does if λ is regular in the sense that for any compact C, λ(C) is the inf of λ(D) for compact sets D containing C in their interiors.

References

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  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010