  The Rademacher functions are investigated in the Morrey spaces M(p,w) on
[0,1] for 1 \le p <\infty and weight w being a quasi-concave function. They
span l_2 space in M(p,w) if and only if the weight w is smaller than the
function log_2^{-1/2}(2/t) on (0,1). Moreover, if 1 < p < \infty the Rademacher
sunspace R_p is complemented in M(p,w) if and only if it is isomorphic to l_2.
However, the Rademacher subspace is not complemented in M(1,w) for any
quasi-concave weight w. In the last part of the paper geometric structure of
Rademacher subspaces in Morrey spaces M(p,w) is described. It turns out that
for any infinite-dimensional subspace X of R_p the following alternative holds:
either X is isomorphic to l_2 or X contains a subspace which is isomorphic to
c_0 and is complemented in R_p.
