Prove Lemma 10.8. Data from lemma 10.8 On the measure space (X, A,p) let us M(A). The

Prove Lemma 10.8.

Data from lemma 10.8

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On the measure space (X, A,p) let us M(A). The set function + 1 ud = [1 sudu, AE A V: A+ is a measure on (X, A). It is called the measure with density (function) u with respect to . We write v=up or du=udp. Proof This is left as an exercise. If v has a density w.r.t. , one writes traditionally du/du for the density function. This notation is to be understood in a purely symbolical way; this is motivated by the well-known fundamental theorem of integral and differential calculus (for Riemann integrals) du U(x) - U(a) = f* u(v)dy dx Using our notation X=A and X(dx)=dx this becomes d(ux) dU dx dx If u > 0 is positive, then it induces a measure v: = = U. =U. v[a, b) = U(b) - U(a) = = funsrud ud (see Problem 6.1). A more advanced discussion of derivatives can be found in Chapter 25, Theorem 25.20 and Appendix I, items 1.16-1.19.