What I mean by random infinite binary sequence is an infinite sequence of $0$'s and $1$'s with probability of occurrence in this sequence equal to $1/2$ (all digits being equally likely).
How is it defined and what are its properties? Are there many such (different) sequences or is it just one?
Can we say that the Champernowne constant in base two $C_2 = 0.11011100101110111…$ (actually the digits after the "$0.$") is such a sequence?
There are several different notions of "randomness", the most generally used one is the Martin-Lof random. It has 3 equivalent definitions. Intuitively, a random sequence is hard to compress, hard to describe, and hard to predict the next digit, which is where the 3 definitions come from.
Champernowne constant is NOT a random sequence，it is a computable real, no computable real is random.