CHAPTER 1
Probability, measure and integration
This chapter is devoted to the mathematical foundations of probability theory.
Section 1.1 introduces the basic measure theory framework, namely, the probability space and the -fields of events in it. The nex
1.1. PROBABILITY SPACES AND -FIELDS
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When is uncountable such a strategy as in Example 1.1.4 will no longer work.
The problem is that if we take p = P(cfw_) > 0 for uncountably many values of
, we shall end up with P() = . Of course we may define everyth
1.2. RANDOM VARIABLES AND THEIR EXPECTATION
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(, F ), denote by (Xk , k n) the smallest -field F such that Xk (), k = 1, . . . , n
are measurable on (, F ). That is, (Xk , k n) is the smallest -field containing
(Xk ) for k = 1, . . . , n.
Remark. One cou
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1. PROBABILITY, MEASURE AND INTEGRATION
Definition 1.1.2. A pair (, F ) with F a -field of subsets of is called a measurable space. Given a measurable space, a probability measure P is a function
P : F [0, 1], having the following properties:
(a) 0 P(A)
1.2. RANDOM VARIABLES AND THEIR EXPECTATION
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1.2.1 and study some of their properties in Subsection 1.2.2. Taking advantage
of these we define the mathematical expectation in Subsection 1.2.3 as the corresponding Lebesgue integral and relate it to the m
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1. PROBABILITY, MEASURE AND INTEGRATION
(a) Show that if g is a continuous function then for each a R the set
cfw_x : g(x) a is closed. Alternatively, you may show instead that cfw_x :
g(x) < a is an open set for each a R.
(b) Use whatever you opted to