the expectation notation

Question

Hi I am going through some definition and proofs of Uniform integrability. I am just confused with these two notations: E(|$X_\alpha$|;|$X_\alpha$|>K) and E(|$X_\alpha$| $|$ |$X_\alpha$|>K); Are they equal? To my understanding, they seems both integrate in the subset {X($\omega$)|$X_\alpha$|>K}? Thanks

Answer 1

The two notations are not necessarily the same, but it really depends on which textbook you're referring to. From my understanding, the notation $E(X|A)$ is reserved for conditional expectation, i.e., $=E(X1_A)/P(A)$, while the notation $E(X;A) = E(X1_A)$ is used in Durrett