A counterexample to the hypothesis on the $H^\infty$ to be dense in the space BMO

In the probability space $(\Omega,\mathscr F,\mathbf P)$ we consider a discrete increasing family of $\sigma$-fields $(\mathscr F_n)$ satisfying special conditions. By means of the norm (which is equivalent to that of the space BMO of martingales) we obtain an example of a martingale which belongs to BMO but cannot be approximated (in the BMO-norm) by elements of $H^\infty$.