be a measure space. Then the simple functions aredense in L^{p}

(Ω)

.

Proof: Recall that a function, f, having values in ℝ can be written in the form f = f^{+}− f^{−}
where

f+ = max (0,f ),f − = max (0,− f) .

Therefore, an arbitrary complex valued function, f is of the form

( )
f = Re f+ − Ref− + i Im f+ − Im f− .

If each of these nonnegative functions is approximated by a simple function, it follows f is also
approximated by a simple function. Therefore, there is no loss of generality in assuming at the outset that
f ≥ 0.

Since f is measurable, Theorem 19.1.6 implies there is an increasing sequence of simple functions, {s_{n}},
converging pointwise to f(x).Now

|f(x)− sn(x)| ≤ |f(x)|.

By the Dominated Convergence theorem,

∫
0 = lim |f(x)− s (x)|pdμ.
n→∞ n

Thus simple functions are dense in L^{p}. ■

Recall that for Ωa topological space, C_{c}(Ω)is the space of continuous functions with compact support in
Ω.Also recall the following definition.

Definition 22.2.2Let (Ω,S,μ) be a measure space and suppose

(Ω,τ)

is also a topological space. Then
(Ω,S,μ) is called a regularmeasure space if the σ algebra of Borel sets is contained in S and for allE ∈S,

μ(E) = inf{μ(V) : V ⊇ E and V open }

and if μ

(E )

< ∞,

μ (E ) = sup{μ(K ) : K ⊆ E and K is compact }

and μ

(K )

< ∞ for any compact set, K.

For example Lebesgue measure is an example of such a measure. More generally these measures are
often referred to as Radon measures. The following useful lemma is stated here for convenience. It is
Theorem 21.0.3 on Page 1552.

Lemma 22.2.3Let Ω be a metric space in which the closed balls are compact and let K be acompact subset of V , an open set. Then there exists a continuous function f : Ω → [0,1] such thatf(x) = 1 for all x ∈ K andspt(f) is a compact subset of V . That is, K ≺ f ≺ V.

It is not necessary to be in a metric space to do this. You can accomplish the same thing using
Urysohn’s lemma.

Theorem 22.2.4Let (Ω,S,μ) be a regular measure space as in Definition 22.2.2where theconclusion of Lemma 22.2.3holds. Then C_{c}(Ω) is dense in L^{p}(Ω).

Proof: First consider a measurable set, E where μ

(E )

< ∞. Let K ⊆ E ⊆ V where μ

(V ∖K )

< ε. Now
let K ≺ h ≺ V. Then

∫ ∫
|h − XE|pdμ ≤ XVp∖Kd μ = μ(V ∖K ) < ε.

It follows that for each s a simple function in L^{p}

(Ω)

, there exists h ∈ C_{c}

(Ω)

such that

||s − h||

_{p}< ε. This
is because if

∑m
s(x) = ciXEi (x)
i=1

is a simple function in L^{p} where the c_{i} are the distinct nonzero values of s each μ

(Ei)

< ∞ since otherwise
s

∕∈

L^{p} due to the inequality

∫
|s|p dμ ≥ |ci|pμ (Ei).

By Theorem 22.2.1, simple functions are dense in L^{p}