Why perfectly secrecy needs the key space to be as large as the message space?
I think the definition (1)
$\Pr[M=m\mathrel|C=c]= Pr[M=m]$ still holds.
Let $M(c)$ be the set of messages that can be decrypted from $c$. We know that $|M(c)| \leq |K| < |M|$ where $K$ and $M$ are the key and message space, and we can infer that some message, say $m'$ can never be the plaintext of $c$, but how does that violate the definition (1) ?