Multiple access channels with arbitrarily correlated sources

Let
\{(U_{i},V_{i})\}_{i=1}^{n}
be a source of independent identically distributed (i.i.d.) discrete random variables with joint probability mass function
p(u,v)
and common part
w=f(u)=g(v)
in the sense of Witsenhausen, Gacs, and Körner. It is shown that such a source can be sent with arbitrarily small probability of error over a multiple access channel (MAC)
\{\cal X_{1} \times \cal X_{2},\cal Y,p(y|x_{1},x_{2})\},
with allowed codes
\{x_{l}(u), x_{2}(v)\}
if there exist probability mass functions
p(s), p(x_{1}|s,u),p(x_{2}|s,v)
, such that
H(U|V)<I(X_{1}; Y|X_{2},V,S),
H(V|U )<I(X_{2};Y|X_{1},U,S),
H(U,V|W)<I(X_{1},X_{2};Y|W,S),
H(U,V)<I(X_{1},X_{2};Y),
\mbox{where}
p(s,u,v,x_{1},x_{2},y), Xl, X2, y)=p(s)p(u,v)p(x_{1}|u,s)p(x_{2}|v,s)p(y|x_{1},x_{2}).
lifts region includes the multiple access channel region and the Slepian-Wolf data compression region as special cases.