First is a significant result on approximating with simple functions in L^{p}.

Theorem 7.4.1Let f ∈ L^{p}

(Ω )

for p ≥ 1. Then for each ε > 0 there is a simple function s suchthat

∥f − s∥ ≤ ε
p

Proof: It suffices to consider the case where f ≥ 0 because you can then apply what is shown to the
positive and negative parts of the real and imaginary parts of f to get the general case. Thus, suppose
f ≥ 0 and in L^{p}

(Ω)

. By Theorem 5.1.9, there exists a sequence of simple functions increasing to f. Then

|f (ω )− sn(ω)|

^{p}≤

|f (ω)|

^{p}. This is a suitable dominating function. Then by the dominated convergence
theorem,

∫
0 = lim |f (ω)− sn (ω )|pdμ
n→ ∞ Ω

which establishes the desired conclusion unless p = ∞.

Use Proposition 7.3.9 to get a set of measure zero N such that off this set,

|f (ω)|

≤

∥f∥

_{∞}. Then
consider fX_{NC}. It is a measurable and bounded function so by Theorem 5.1.9, there is an increasing
sequence of simple functions

{sn}

converging uniformly to this function. Hence, for n large enough,

∥f − sn∥

_{∞}< ε. ■

Theorem 7.4.2Let μ be a regular Borel measure on ℝ^{n}and f ∈ L^{p}

(ℝn )

. Then for eachp ≥ 1,p≠∞, there exists g a continuous function which is zero off a compact set such that

∥f − g∥

_{p}< ε.

Proof:Without loss of generality, assume f ≥ 0. First suppose that f is 0 off some ball B

(0,R)

. There
exists a simple function 0 ≤ s ≤ f such that ∫

|f − s|

^{p}dμ <

(ε∕2)

^{p}. Thus it suffices to show the existence
of a continuous function h which is zero off a compact set which satisfies

(∫ p )
|h− s| dμ

^{1∕p}< ε∕2.
Let

m∑
s(x) = ciXEi (x) ,Ei ⊆ B (0,R )
i=1

where E_{i} is in ℱ_{p}. Thus each E_{i} is bounded. By regularity, there exist compact sets K_{i} and open sets V_{i}
such that K_{i}⊆ E_{i}⊆ V_{i}⊆ B