What is perfect secrecy? I read some similar questions like Simply put, what does “perfect secrecy” mean?
 (This one defines perfect secrecy as ciphertext conveys no information about the content of the plaintext.

Now, problem 2.3 on this assignment asks:


  show that perfect secrecy of $(GEN, ENC, DEC)$ implies $$\Pr [ENC_k(m) = c] = \Pr [ENC_k(m') = c]$$


Isn't the above equation saying that ciphertext $c$ gives no information whether it's $m$ or $m'$? 

What is the exact definition of perfect secrecy then?

P.S. the assignment link above contains the solution. In the solution, I don't understand where the definition of perfect secrecy is used, nor what it is.
The Lindell and Katz, in their book, give the definition as follows;


  An encryption scheme $(Gen,Enc,Dec)$ over a message space
  $\mathcal{M}$ is perfectly secret if for every probability
  distribution of over $\mathcal{M}$, every every message
  $m\in\mathcal{M}$, and every ciphertext $c \in \mathcal{C}$ for which
  $Pr[C=c]&gt;0$;
  
  $$Pr[M=m|C=c] = Pr[M=m]$$


We can view this definition as the distribution over messages and ciphertext are independent.



The proof used this definition and the equivalent definition of perfect secrecy

$Pr[C=c|M=m] = Pr[C=c]$

and, this can be said, the ciphertext reveals no information about the plaintext other than the size.



On your proof;

On the third P, on the numerator and denominator; 

$Pr[Enc_k(m_1)=c] = Pr[Enc_k(M) =c | M=m_1]$