Review: The Economist is an international weekly journal written should you percentage an unusual curiosity in being good and generally proficient. each one factor explores the shut hyperlinks among family and overseas matters, enterprise, politics, finance, present affairs, technological know-how, expertise and the humanities.

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**Example text**

F) . 6. 7 (Radon-Nikodym) . P(A) = l Z(w) dlP'(w) for every A E :F. 1 . 2) . Here !? is the set of all possible outcomes of a random experiment, :F is the collection of subsets of n whose probability is defined, and IP' is a function mapping :F to [0, 1] . ) = 1 and countable additivity: the probability of a union of disjoint sets is the sum of the probabilities of the individual sets. The collection of sets :F in the preceding paragraph is a a-algebra, which means that 0 belongs to :F, the complement of every set in :F is also in :F, and the union of any sequence of sets in :F is also in :F.

2, the whole sequence must have probability zero. Therefore, IP'{w E [0, 1] ; w is rational} = 0. Since IP'[O, 1] = 1, the probability of the set of irrational numbers in [0, 1] must be 1. We conclude that lEX = 1. The idea behind this example is that if we choose a number from [0, 1] according to the uniform distribution, then with probability one the number chosen will be irrational. Therefore, the random variable X is almost surely equal to 1, and hence its expected value equals 1. As a practical matter, of course, almost any algorithm we devise for generating a random number in [0, 1] will generate a rational number.

The point is that lEX can be infinite, even though X is finite almost surely. 0 The following theorem provides another common situation in which we are assured that the limit of the integrals of a sequence of functions is the integral of the limiting function. Let XI , x2 , . . be a sequence variables converging almost surely to a random variable X . If there is another random variable Y such that lEY < oo and I Xn l :::; Y almost surely for every n, then limoo EXn = EX . n-+ Let /I , h, . . be a sequence of Borel-measurable functions on JR.

### The Economist (14 May 2016)

by Edward

4.3