In a discrete space, everything is centered around the function "P". Every event has a probability associated with it, and random variables simply map outcomes to numbers. P(X = x) is basically just a weird way of saying P({e | X(e) = x}). From "P", people can define f(x) = P(X = x), g(x|y) = P(X = x, Y = y) / P(Y = y), and so on.
But in the continuous space it seems like everything is reversed! Whenever I look in a textbook, authors will just define a pdf (e.g. f(x,y)) and then derive "P" from it! (e.g. P((X,Y) $\in$ A) = $\int\int$f(x,y). How can this be?
A related question would be how random variables work in continuous spaces, too. Do they still map events to probabilities? Surprisingly, none of my textbooks talk about this...
Hi Michael Chernick, thank you for directing me to those other posts, but I just read the accepted answer in the first one, and it only talked about discrete variables. The second post supposes that I know what sigma algebras are, which I've never heard of before. The accepted answer talks about it, but also doesn't really answer why textbooks go from pdf -> P()
