So I am a math major who is trying to learn some crypto. However I have some difficulties with some of the probability definitions that are assumed in the cryptography book that I am using at the moment. So here it goes:
Def(perfect security): Let $(E,D)$ be a Shannon cipher defined over $(K,M,C)$ where all these are finite sets and are respectively the key space, message space, and ciphertext space. Now consider the probabilistic experiment in which the random variable $k$ is uniformly distributed over $K$. If for all $m_0$, $m_1 \in M$, and all $c\in C$, we have $P(E(k,m_0)=c) = P(E(k,m_1)=c)$, then we say $(E,D)$ is perfectly secure.
As I understand it, the author assumes that we have a probability space on $K$, where the sigma-algebra is taken to be the power set of $K$, and $P$ is the probability measure defined on the power set of $K$. What I don't understand is what does random variable mean here. If I recall correctly, a random variable in probability theory is defined to be a measurable function from(in our case) $K$ to the real numbers. But I do not think the author is using this definition of random variable here. It would be great if anyone could clarify this part of the definition and also clarify what does $k$ uniformly distributed mean rigorously as well. I thought it means probability of each singleton outcome in $K$ is $1/|K|$. I might be wrong though. Thanks.