An achievable rate region for the broadcast channel

Let
p(y_1, y_2 \mid x)
denote a discrete memoryless channel with a single source
X
and two independent receivers
Y_1
and
Y_2
. We exhibit an achievable region of rates
(R_{11},R_{12},R_{22})
at which independent information can be sent, respectively, to receiver 1, to both receivers 1 and 2, and to receiver 2. The achievability of the region is shown by using a version of the asymptotic equipartition property involving many simultaneous "typicality" constraints. These results immediately generalize to yield an achievable rate region for the
m
-sender
n
-receiver channel in terms of standard mutual information quantities.