In perfectly secret scheme, if key and message space are uniformly distributed, is ciphertext space always uniform as well? Following up on a similar question, Does perfect secrecy imply uniform ciphertext distribution?, the answer seems to be that in a perfectly secret encryption scheme, the distribution of the ciphertext space is not necessarily uniform because the message space is not always uniformly distributed (although the key space is assumed to be uniform). 

Question: If we say that the message space and key space are uniformly distributed, this seems to me to imply that the ciphertext space must also be uniform. How would one go about proving this fact?
Question: If we say that the message space and key space are uniformly
  distributed, this seems to me to imply that the ciphertext space must
  also be uniform. How would one go about proving this fact?


You don't because this statement is false.
Here's a simple counter-example:

Let $k,m\in\{0,1\}$ be a key and a message bit, both chosen uniformly at random. Then let the ciphertext $c=m\oplus k\parallel 0$, that is you do the standard OTP encryption and append a zero bit. Obviously this scheme is perfectly secret because the OTP is and the zero bit is completely independent of the key and the message. Also obviously the ciphertext doesn't have a uniform distribution from the set $\{00,01,10,11\}$ because $11$ cannot occur.

A more interesting (new!) question though would be though whether for each length-preserving perfectly-secret encryption scheme the ciphertext follows a uniform distribution.