  With the aim of extracting information on the shift of the $\omega$ mass in
the nuclear medium we analyze data obtained at ELSA from where claims for
evidence of a mass shift of the $\omega$ have been made. We develop a Monte
Carlo simulation code which takes into account the possible reactions in the
experimental set up of $(\gamma ~ A \to \pi^0 \gamma ~X)$ in the vicinity of
the $\omega$ production region with subsequent $\omega \to \pi^0 \gamma$ decay.
We compare our results with experiment for the distribution of $\pi^0 \gamma$
invariant masses and conclude that the distribution is compatible with an
enlarged $\omega$ width of about 90 MeV at nuclear matter density and no shift
in the mass. This change in the width would be compatible with the preliminary
results obtained from the transparency ratio in the A dependence of $\omega$
production. The discrepancy of the present conclusions with former claims of an
evidence for a shift of the $\omega$ mass stem from a different choice of
background which is discussed in the paper.
