If X and Y are two identically distributed integrable r.v.s then For any constant c . Now,
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For any constant c.
Now, let g be nonnegative. Then there exist 0 £ gn(x) simple †‘ g(x); i.e.,
Which implies that 0 £
Simple †‘ g(X) as n†’¥ where Ani = X€“1(Bni). Then
Whereas
For all n, by the previous step. Hence
Finally, for any g, write g(x) = g+ (x) €“ g€“(x), which implies g(X) = g+(X) €“ g€“ (X). Now, if fWg (X)d P exists, it then follows that either
Or both. Since
And
By the pervious step, it follows that either
Or both, respectively, Thus, fÂg(x)d Px exists and
Likewise, the existence of fÂg (x)d Px implies the existence of fWg (X)d P and their equality.
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Related Book For
An Introduction to Measure Theoretic Probability
ISBN: 978-0128000427
2nd edition
Authors: George G. Roussas
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