  The real part of $H^\infty(\bT)$ is not dense in $L^\infty_{\tR}(\bT)$. The
John-Nirenberg theorem in combination with the Helson-Szeg\"o theorem and the
Hunt Muckenhaupt Wheeden theorem has been used to determine whether $f\in
L^\infty_{\tR}(\bT)$ can be approximated by $\Re H^\infty(\bT)$ or not:
$\dist(f,\Re H^\infty)=0$ if and only if for every $\e>0$ there exists $\l_0>0$
so that for $\l>\l_0$ and any interval $I\sbe \bT$. $$|\{x\in I:|\tilde
f-(\tilde f)_I|>\l\}|\le |I|e^{-\l/ \e},$$ where $\tilde f$ denotes the Hilbert
transform of $f$. See [G] p. 259. This result is contrasted by the following
  \begin{theor} Let $f\in L^\infty_{\tR}$ and $\e>0$. Then there is a function
$g\in H^\infty(\bT)$ and a set $E\sb \bT$ so that $|\bT\sm E|<\e$ and $$f=\Re
g\quad\mbox{ on } E.$$ \end{theor}
  This theorem is best regarded as a corollary to Men'shov's correction
theorem. For the classical proof of Men'shov's theorem see [Ba, Ch VI \S
1-\S4].
  Simple proofs of Men'shov's theorem -- together with significant extensions
-- have been obtained by S.V. Khruschev in [Kh] and S.V. Kislyakov in [K1],
[K2] and [K3].
  In [S] C. Sundberg used $\bar\pa$-techniques (in particular [G, Theorem
VIII.1. gave a proof of Theorem 1 that does not mention Men'shov's theorem.
  The purpose of this paper is to use a Marcinkiewicz decomposition on
Holomorphic Martingales to give another proof of Theorem 1. In this way we
avoid uniformly convergent Fourier series as well as $\bar\pa$-techniques.
