If m: message, M: message space, k: key, K: keyspace, c: cipher, C: cipher space and $E_k$: encryption function, such that $E_k(m) = c,\ m,m^* \in M,\ k\in K,\ c\in C.$
Then, what do the following notations actually mean in plain English? \begin{eqnarray*}P[m=m^* | E_k(m) = c] = \frac1{|M|}\end{eqnarray*}
\begin{eqnarray*}P[m=m^*] = P[E_k(m) = E_k(m^*)]\end{eqnarray*}
I am trying the notation to match it with https://www.lri.fr/~fmartignon/documenti/systemesecurite/3-OneTimePad.pdf and https://www.udacity.com/course/viewer#!/c-cs387/l-48735205/e-48734451/m-48738141 to understand it.
The first reads, "given that the encryption function $E_k$ using key $k$ applied to message $m$ returns the cipher $c$, the probability that two messages, $m$ and $m^∗$, are equal is $\frac{1}{|M|}$."
The second line reads "The probability that two messages, $m$ and $m^∗$ are the same is equal to the probability that their encrypted messages, $E_k(m)$ and $E_k(m^∗)$, are the same."
In particular, it is worth noting what the symbols used in each statement mean. The symbol $P[A]$ represents the probability measure which assigns a value between (inclusive) zero and one to the event, $A$, enclosed in the brackets.
When there is a vertical bar inside of the brackets however, it is a conditional probability. $P[A|B] := \frac{P[A\cap B]}{P[B]}$, and is read aloud as "the probability of $A$ given $B$" and has the interpretation of "supposing that we know ahead of time that $B$ is true/hashappened, the probability that $A$ is also true is ..."
Here also we have another remark to be made about notation, often times you will see $P[A]$ where $A$ is an event which is defined elsewhere. In our case, we see the event being defined inside of the brackets. $P[m=m^*]$ is the probability of the event that $m=m^*$, i.e. the probability that the two messages $m$ and $m^*$ are actually the same.