Struggling to read the notation for the definition in the pink box. What is "U" and what is "o" ?It would be helpful to write out the conventional way to read it as denoted by the notation. Below are the math notations I understand. The ones I don't know I highlighted with a "?" X = random variableR = all real numbers(Omega) = sample spaceC = subset-> = "maps to ": = "such that"g = ? (Don't know)"o" = ? (Don't know )U = ? (Don't know)

event {X =1} should be understood to mean the set of states (W E S : A (W) - IS. There are two ways one can apply functions to random variables. The first is to use the value of X(w) as input into another function g, with the result being another random variable. The second is to operate on the function X(. ) itself, returning a value characterizing some feature of the function. We provide two formal definitions that we will delineate notationally. Definition 1.2.2. Function of a Random Variable Let g : U - R be a function, where X(2) C UC R. Then, if go X : 2- R is a random variable, we say that g is a function of X, and write g(X) to denote the random variable go X. 15 14 This regularity condition is necessary and sufficient to ensure that X is a measurable function