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Let X be a set A family F of subsets of X is a -algebra if and only if it has the following properties: (i) The empty set is in F (ii) If A is in F , then so is the complement of A (iii) If A1 , A2 , A3 , is a sequence in F , then their (countable) union is also in F From (i) and (ii) it follows that X is in F ; from (ii) and (iii) it follows that the -algebra is also closed under countable intersections -algebras are mainly used to de ne measures on X, as are de ned in the next section An ordered pair (X , F ), where X is a set and F is a -algebra over X, is called a measurable space



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1 The family consisting only of the empty set and X is a -algebra over X, the so-called trivial -algebra Another -algebra over X is given by the power set of X, that is, the set P(X ) of all subsets of X 2 If {Fa } is a family of -algebras over X, then the intersection of all Fa is also a -algebra over X 3 If U is an arbitrary family of subsets of X, then we can form a special -algebra from U , called the -algebra generated by U We denote it by (U ) and de ne it as follows First note that there is a -algebra over X that contains U , namely, the power set of X Let be the family of all -algebras over X that contain U (ie, a -algebra F over X is in if and only if U is a subset





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new funding formula should be introduced based 50 percent on population and 50 percent on economic output (on a PPP basis) Assume a larger UN budget set at $10 billion a dramatic increase from its current $2 billion level, but still fairly low Under this new arrangement the United States (with 43 percent of the world s population) would pay approximately $228 million in population dues plus $1066 billion in economic dues (to account for its 218 percent share of world output) That makes for a total payment of less than $13 billion, more than double today s contributions but less than 11 percent of the total UN bill versus nearly 22 percent today This is but a small part of the United States aggregate defense/intelligence/diplomatic spending that is currently nearing $800 billion

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of F ) Then we de ne (U ) to be the intersection of all -algebras in (U ) is then the smallest -algebra over X that contains U 4 This leads to the most important example: the Borel algebra over any topological space is the -algebra generated by the open sets (or, equivalently, by the closed sets) Note that this -algebra is not, in general, the whole power set 5 On the Euclidean space Rn , another -algebra is of importance: that of all Lebesgue measurable sets This -algebra contains more sets than the Borel algebra on Rn and is preferred in integration theory

A measure is a function de ned on a -algebra F over a set X and taking values in [0, ) such that the following properties are satis ed: 1 The empty set has measure zero: ( ) = 0 (A1)

Address range: $000000, $000002, ... , $001FFE Similarly, the memory for the odd 2732, even 61 16, and odd 61 16 can be determined as follows: 2732odd

.

2 Countable additivity or -additivity: if E 1 , E 2 , E 3 , is a countable sequence of pairwise disjoint sets in F , the measure of the union of all the E i is equal to the sum of the measures of each E i :

(E i )

s ( s )

(A2)

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