Cover and Thomas' Remark on Average Probability of Error for an $(M, n)$ code

Question

I am confused by one remark on the average probability of error for $(M, n)$ code given in Cover and Thomas' book "Elements of Information Theory".

They said $P_e^{(n)}$ is a probability of error only if the message is chosen uniformly over the message set $\{1, 2, ..., 2^M\}$. Isn't the message set $\{1, 2, ..., M\}$? Where does this $2^M$ come from? Is this an errata?

Thanks to everyone who's trying to help out.

Here are the definitions of $(M, n)$ code and the average probability of error $P_e^{(n)}$ given in the book. Also the exact statement I have a problem with (highlighted):

Answer 1

Yes, this is an error. Well spotted.

The correct paragraph would be:

over the message set $\{1,2,...,M\}$