Probability, Random Processes, and Ergodic Properties by Robert M. Gray

Author:Robert M. Gray [Gray, Robert M.]
Language: eng
Format: epub, pdf
Published: 2011-02-19T05:00:00+00:00

5.2. MEASUREMENTS AND EVENTS

97

where the F (i) are disjoint, and hence since gn ∈ M(σ(f )) that g−1(r

n

i) = F (i) ∈ σ(f ).

Since σ(f ) = f −1(B(A)), there are disjoint sets Q(i) ∈ B(A) such that F (i) = f−1(Q(i)). Define the function hn : A →

by

M

hn(a) =

ri1Q(i)(a)

i=1

and

M

M

hn(f (ω))

=

ri1Q(i)(f(ω)) = ri1f−1(Q(i))(ω)

i=1

i=1

M

=

ri1F (i)(ω) = gn(ω). i=1

This proves the result for simple functions. By construction we have that g(ω) = limn→∞ gn(ω) =

limn→∞ hn(f(ω)) where, in particular, the right-most limit exists for all ω ∈ Ω. Define the function h(a) = limn→∞ hn(a) where the limit exists and 0 otherwise. Then g(ω) = limn→∞ hn(f(ω)) =

h(f (ω)), completing the proof.

2

Thus far we have developed the properties of σ-fields induced by random variables and of classes of functions measurable with respect to σ-fields. The idea of a σ-field induced by a single random variable is easily generalized to random vectors and sequences. We wish, however, to consider the more general case of a σ-field induced by a possibly uncountable class of measurements. Then we will have associated with each class of measurements a natural σ-field and with each σ-field a natural class of measurements. Toward this end, given a class of measurements M, define σ(M) as the smallest σ-field with respect to which all of the measurements in M are measurable. Since any σ-field satisfying this condition must contain all of the σ(f ) for f ∈ M and hence must contain the σ-field induced by all of these sets and since this latter collection is a σ-field, σ(M) = σ(

σ(f )).

f ∈M

The following lemma collects some simple relations among σ-fields induced by measurements and classes of measurements induced by σ-fields.

Lemma 5.2.2 Given a class of measurements M, then

M ⊂ M(σ(M)).

Given a collection of events G, then

G ⊂ σ(M(σ(G))).

If G is also a σ-field, then

G = σ(M(G)),

that is, G is the smallest σ-field with respect to which all G-measurable functions are measurable. If G is a σ-field and I(G) = {all 1G, G ∈ G} is the collection of all indicator functions of events in G, then

G = σ(I(G)),

that is, the smallest σ-field induced by indicator functions of sets in G is the same as that induced by all functions measurable with respect to G.

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