362 CHAPTER 12. THE Lp SPACES

It was just shown that ∥ f∥∞

is the smallest constant such that | f (x)| ≤ ∥ f∥∞

a.e. Henceif f ,g ∈ L∞ (Ω) ,| f (x)+g(x)| ≤ | f (x)|+ |g(x)| ≤ ∥ f∥

∞+ ∥g∥

∞a.e. and so by definition

∥ f +g∥∞≤ ∥ f∥

∞+∥g∥

∞.

Next suppose you have a Cauchy sequence in L∞ (Ω) { fn} . Let | fn (x)− fm (x)| <∥ fn− fm∥∞

for x /∈ Enm, µ (Enm) = 0 and let | fn (x)| ≤ ∥ fn∥∞for x /∈ En, µ (En) = 0. Then

let E ≡ ∪nEn ∪∪m,nEnm. It follows that for x /∈ E, limn→∞ fn (x) exists. Let f (x) be thislimit for x /∈ E and let f (x) = 0 on E. Also |∥ fn∥∞

−∥ fm∥∞| ≤ ∥ fn− fm∥∞

since ∥·∥∞

is anorm. Therefore, for x /∈ E,| f (x)|= limn→∞ | fn (x)| ≤ limn→∞ ∥ fn∥∞

≡C so f ∈ L∞ (Ω,µ).Also, for x /∈ E, | fm (x)− fn (x)| ≤ ∥ fm− fn∥∞

< ε if m > n and n is large enough. There-fore, for such n, letting m→∞, | f (x)− fn (x)| ≤ ε for x /∈ E. It follows that ∥ f − fn∥∞

≤ ε

if n large enough and so by definition, limn→∞ ∥ f − fn∥∞= 0. ■

12.2 Density ConsiderationsTheorem 12.2.1 Let p≥ 1 and let (Ω,S ,µ) be a measure space. Then the simplefunctions are dense in Lp (Ω). In fact, if f ∈ Lp (Ω) , then there is a sequence of simplefunctions {sn} such that |sn| ≤ | f | and ∥ f − sn∥p→ 0.

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+−Re f−+ i(Im f+− Im f−

).

If each of these nonnegative functions is approximated by a simple function, it follows f isalso approximated by a simple function. Approximating each of the positive and negativeparts with simple functions having absolute value less than what is approximated, it wouldfollow that |sn| ≤ 4 | f | and all that is left is to verify that ∥sn− f∥p→ 0 which occurs if ithappens for each of these positive and negative parts of real and imaginary parts.

Now | f (x)− sn(x)| ≤ 5 | f | and so | f (x)− sn(x)|p ≤ 5p | f |p which is in L1. Then by thedominated convergence theorem, 0 = limn→∞

∫| f (x)− sn(x)|pdµ showing that the simple

functions are dense in Lp. ■Note how this observation always holds and requires no assumptions on the measures.Recall that for Ω a topological space, Cc(Ω) is the space of continuous functions with

compact support in Ω. Also recall the following definition.

Definition 12.2.2 Let (Ω,S ,µ) be a measure space and suppose (Ω,τ) is also atopological space (metric space if you like.). Then (Ω,S ,µ) is called a regular measurespace if the σ algebra of Borel sets is contained in S and for all E ∈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.