1. As H imbeds O in X * , there exists a unique x ? O such that H(x) = ( O �(dz) h(z)) h?X * . By the definition of H;O �(dz) h(z)) h?X * ? H(O)
2. Consider the portfolio ? = 1 ? ?x+?y ? ?1 ?x ? ?1 ?y . Note that supp ? = {? ?x+?y , ? x , ? y } ? ?(X) 0 . Then, �?supp ? ?(�)?(�) = ?(? ?x+?y ) ? ??(? x ) ? ??(? y ) = p(?x + ?y) ? ?p(x) ? ?p(y) < 0 and �?supp ? ?(�)m � = ?x + ?y ? ?x ? ?y = 0, which contradicts the arbitrage-free condition. Suppose p is not increasing. Then there exists x ? X such that x > 0 and p(x) ? 0. If ? = 1 ?x , then �?supp ? ?(�)?(�) = ?(? x ) = p(x) ? 0 and �?supp ? ?(�)m � = x > 0;O �(dz) h(z) for every h ? X * . Set m � = x. Proof of Lemma 8.2 (A) Suppose p is not linear. Then there exist x, y ? X and ?, ? ? such that p(?x+?y) < ?p(x)+?p(y)
3. Grenzwerte: $$\mathop {\lim }\limits_{k \to \infty } \int {f\left( {k,x} \right)} dx$$