whenever this sum converges absolutely.

Intuitively, this result is rather clear, since g(X) takes the value g(x) when X takes the value x, an event which has probability P(X = x). A more formal proof proceeds as follows.

Proof Writing I for the image of X, we have that Y = g(X) has image g(I ). Thus E(Y ) =

X y∈g(I )

yP(Y = y)

=

X y∈g(I )

y X

x∈I : g(x)=y P(X = x) by (2.25)

=

X x∈I g(x)P(X = x)