A Mixing system is defined as a dynamical system $(\Omega,\phi^t,\mu)$ for which the following relations holds $$ 1)\qquad\lim_{t\rightarrow\infty} \mu(\phi^{-t}(A)\cap B)=\mu(A)\mu(B); $$ $$ 2)\qquad\lim_{t\rightarrow\infty} \langle F\circ\phi^{-t}G\rangle_\mu=\langle F\rangle_\mu\langle G\rangle_\mu. $$ It is easy to show that $2\Rightarrow1$ using as $F=\chi_A$ and $G=\chi_B$. The $1\Rightarrow2$ implication follow from density of simple functions $(\mathcal{F}=\sum_i a_i\chi_{A_i})$ on $\mathcal{L}_2(\Omega,\mu)$, indeed $\mu(A)=\langle \chi_A\rangle_\mu$.
Now, what should I do if I want to show the second implication by making explicit use of density argument, i.e $||F-\mathcal{F}||_2<\epsilon$, $\epsilon>0$?
I've already used the density argument on $\mathcal{L}_1$ when i proved that the non resoanting translation on tori are ergodic, ($||\bar{f}-\langle f\rangle_\mu||_1<4\epsilon$, using the trigonometric polynomial $P_N(\varphi)$ and the fact that $||f-P_N||_1<\epsilon$ $\forall f\in\mathcal{L}_1$), so i know that i have to show that exist a positive constant $\epsilon>0$ such that $$ |\langle F\circ\phi^{-t}G\rangle_\mu-\langle F\rangle_\mu\langle G\rangle_\mu|<c\epsilon $$ but i am stuck. Thanks in advance for any hint!