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Additional info for An Introduction to Statistical Signal Processing last edition
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. The fourth axiom can be viewed as an extra technical condition that must be included in order to get various limits to behave. 4, will be shown to have an alternative form in terms of explicit limits, a form providing an important continuity property of probability. Also as in the event space properties, the fourth axiom implies the third.
3. PROBABILITY SPACES 25 that is, the whole sample space considered as a set must be in F; that is, it must be an event. ” A few words about the different nature of membership in Ω and F is in order. If the set F is a subset of Ω, then we write F ⊂ Ω. If the subset F is also in the event space, then we write F ∈ F. Thus we use set inclusion when considering F as a subset of an abstract space, and element inclusion when considering F as a member of the event space and hence as an event. Alternatively, the elements of Ω are points, and a collection of these points is a subset of Ω; but the elements of F are sets — subsets of Ω, — and not points.
The moral of this discussion is that the product sigma-field for spaces of sequences and waveforms must contain (but not consist exclusively of) all sets that are described by requiring that the outputs of coordinates for a finite number of events lie in sets in the one-dimensional event space F. We shall further explore such product event spaces when considering random processes, but the key points remain 1. a product event space is a sigma-field, and 2. it contains all “one-dimensional events” consisting of subsets of the product sample space formed by grouping together all vectors or sequences or waveforms having a single fixed coordinate lying in a onedimensional event.