Capacity of symmetric finite input and continuous output channels

In this work we study channels with finite input alphabet and continuous output alphabet and we give step-by-step proofs on two results: 1.-The capacity of the channel can be computed by finding a random variable over the inputs in such a way that the Kullback-Leibler distance D(pY_|xi (y)∥kpY (y)) is constant for any x_i of the input alphabet. 2.-The capacity of the symmetric channel can be computed by making the random variable over the inputs be uniform, i.e., probability pX(x_i) constant for any input x_i.