Real and Complex Analysis (Rudin)

I!-SPACES 69
The simple functions play an interesting role in I!(p.):
3.13 Theorem Let S be the class of all complex, measurable, simple functions
on X such that
If 1 :s;; p < 00, then S is dense in I!(p.).
p.({x: s(x) #= OJ) < 00. (1)
PROOF First, it is clear that S c: I!(p.). Suppose f ~ O,f E I!(p.), and let {sn} be
as in Theorem 1.17. Since 0 :s;; Sn :s;;f, we have sn E I!(p.), hence sn E S. Since
If - Sn IP :s;;fP, the dominated convergence theorem shows that
IIf - sn IIp-+ 0 as n-+ 00. Thus f is in the I!-closure of S. The general case
(fcomplex) follows from this.
IIII
Approximation by Continuous Functions
So far we have considered I!(p.) on any measure space. Now let X be a locally
compact Hausdorff space, and let p. be a measure on a a-algebra ro1 in X, with
the properties stated in Theorem 2.14. For example, X might be Rk, and p. might
be Lebesgue measure on Rk.
Under these circumstances, we- have the following analogue of Theorem 3.13:
3.14 Theorem For 1 :s;; p < 00, Cc(X) is dense in I!(p.).
PROOF Define S as in Theorem 3.13. If s E Sand € > 0, there exists agE
Cc(X) such that g(x) = s(x) except on a set of measure