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 pro
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 sh
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
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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
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
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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