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**Unformatted text preview: **n encountered in elementary
probability courses, namely if X is a continuous random variable with density fX , then
∞
E(X ) =
xfX (x) dx.
−∞ It turns out that verifying this formula is somewhat more involved than the discrete formula.
As such, we need to take a brief detour into some general function theory.
Some General Function Theory
Suppose that f : X → Y is a function. We are implicitly assuming that f is deﬁned for all
x ∈ X. We call X the domain of f and call Y the codomain of f .
The range of f is the set f (X) = {y ∈ Y : f (x) = y for some x ∈ X}.
Note that f (X) ⊆ Y. If f (X) = Y, then we say that f is onto Y.
Let B ⊆ Y. We deﬁne f −1 (B ) by f −1 (B ) = {x ∈ X : f (x) = y for some y ∈ B } = {f ∈ B } = {x : f (x) ∈ B }.
We call X a topological space if there is a notion of open subsets of X. The Borel σ -algebra
on X, written B (X), is the σ -algebra generated by the open sets of X.
22–2 Let X and Y be topological spaces. A function f : X → Y is called...

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