Quenched Averaged Estimates and the Infinite Volume Polymer Measure

3 Quenched Averaged Estimates and the Infinite Volume Polymer Measure 

In this section we will consider functions real valued functions /( ) defined on the power set of N U 0. Of course such a function may be viewed as an application from 0, equipped with the product topology, to R. 

The main result of this section is self-explanatory  

Theorem 7.7 For every (/ , h) C there exist two positive constants ci 

Proof It suffices to show that E j [/(r)] converges P(dw)-a.s. for every local function. Note that (7.37) implies that E [/(r)] jy is a Cauchy sequence in T (P), and therefore it converges (in T (P)) to a limit random variable that we denote by E [/(t)]. We can therefore consider the limit as N — oo in (7.37) itself and by the Fubini-Tonelli Theorem we obtain 

---