By Gray Robert
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Extra info for An Introduction to Statistical Signal Processing last edition
4 If Fi , i = 1, 2, . . are disjoint, then P ∞ i=1 Fi = ∞ P (Fi ) . 25) i=1 Note that nothing has been said to the effect that probabilities must be sums or integrals, but the first three axioms should be recognizable from the three basic properties of nonnegativity, normalization, and additivity encountered in the simple examples introduced in the introduction to this chapter where probabilities were defined by an integral of a pdf over a set or a sum of a pmf over a set. The axioms capture these properties in a general form and will be seen to include more general constructions, including multidimensional integrals and combinations of integrals and sums.
10) k=1 showing that additivity is equivalent to finite additivity, the extension of the additivity property from two to a finite collection of sets. Since additivity is a special case of finite additivity and it implies finite additivity, the two notions are equivalent and we can use them interchangably. These three properties of nonnegativity, normalization, and additivity are fundamental to the definition of the general notion of probability and will form three of the four axioms needed for a precise development.
While both of the preceding examples can be used to provide event spaces for the special case of Ω = , the real line, neither leads to a useful probability theory in that case. In the next example we consider another event space for the real line that is more useful and, in fact, is used almost always for and higher dimensional Euclidean spaces. First, however, we need to treat the idea of generating an event space from a collection of important events. Intuitively, given a collection of important sets G that we require to be events, the event space σ(G) generated by G is the smallest event space F to which all the sets in G belong.