Geodesic
A counterexample to the hypothesis on the $H^\infty$ to be dense in the space BMO
Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 1, pp. 154-157

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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 $H^\infty$. 
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     author = {I. V. Pavlov},
     title = {A counterexample to the hypothesis on the $H^\infty$ to be dense in the space {BMO}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {154--157},
     publisher = {mathdoc},
     volume = {25},
     number = {1},
     year = {1980},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1980_25_1_a14/}
}
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I. V. Pavlov. A counterexample to the hypothesis on the $H^\infty$ to be dense in the space BMO. Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 1, pp. 154-157. http://geodesic.mathdoc.fr/item/TVP_1980_25_1_a14/