Pound–Rebka experiment

In mathematics, a subset $A$ of a Polish space $X$ is universally measurable if it is measurable with respect to every complete probability measure on $X$ that measures all Borel subsets of $X$. In particular, a universally measurable set of reals is necessarily Lebesgue measurable (see #Finiteness condition) below.

Every analytic set is universally measurable. It follows from projective determinacy, which in turn follows from sufficient large cardinals, that every projective set is universally measurable.

Finiteness condition

The condition that the measure be a probability measure; that is, that the measure of $X$ itself be 1, is less restrictive than it may appear. For example, Lebesgue measure on the reals is not a probability measure, yet every universally measurable set is Lebesgue measurable. To see this, divide the real line into countably many intervals of length 1; say, N₀=[0,1), N₁=[1,2), N₂=[-1,0), N₃=[2,3), N₄=[-2,-1), and so on. Now letting μ be Lebesgue measure, define a new measure ν by

$\nu (A)=\sum _{i=0}^{\infty }{\frac {1}{2^{n+1}}}\mu (A\cap N_{i})$

Then easily ν is a probability measure on the reals, and a set is ν-measurable if and only if it is Lebesgue measurable. More generally a universally measurable set must be measurable with respect to every sigma-finite measure that measures all Borel sets.

Example contrasting with Lebesgue measurability

Suppose $A$ is a subset of Cantor space $2^{\omega }$; that is, $A$ is a set of infinite sequences of zeroes and ones. By putting a binary point before such a sequence, the sequence can be viewed as a real number between 0 and 1 (inclusive), with some unimportant ambiguity. Thus we can think of $A$ as a subset of the interval [0,1], and evaluate its Lebesgue measure. That value is sometimes called the coin-flipping measure of $A$, because it is the probability of producing a sequence of heads and tails that is an element of $A$, upon flipping a fair coin infinitely many times.

Now it follows from the axiom of choice that there are some such $A$ without a well-defined Lebesgue measure (or coin-flipping measure). That is, for such an $A$, the probability that the sequence of flips of a fair coin will wind up in $A$ is not well-defined. This is a pathological property of $A$ that says that $A$ is "very complicated" or "ill-behaved".

From such a set $A$, form a new set $A'$ by performing the following operation on each sequence in $A$: Intersperse a 0 at every even position in the sequence, moving the other bits to make room. Now $A'$ is intuitively no "simpler" or "better-behaved" than $A$. However, the probability that the sequence of flips of a fair coin will wind up in $A'$ is well-defined, for the rather silly reason that the probability is zero (to get into $A'$, the coin must come up tails on every even-numbered flip).

For such a set of sequences to be universally measurable, on the other hand, an arbitrarily biased coin may be used--even one that can "remember" the sequence of flips that has gone before--and the probability that the sequence of its flips ends up in the set, must be well-defined. Thus the $A'$ described above is not universally measurable, because we can test it against a coin that always comes up tails on even-numbered flips, and is fair on odd-numbered flips.

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