Approximation of Lp(X) by simple functions

6 STEVE SHKOLLER

1.6. Approximation of L^p(X) by simple functions.

Lemma 1.23. If p ∈ [1, ∞), then the set of simple functions f =n

i=1a_i1Ei, where each Ei is an element of the σ-algebraA and μ(Ei) <∞, is dense in L^p(X,A, μ).

Proof. If f ∈ L^p, then f is measurable; thus, there exists a sequence{φn}^∞_n=1 of simple functions, such that φn→ f a.e. with

0≤ |φ1| ≤ |φ2| ≤ · · · ≤ |f|, i.e., φ_n approximates f from below.

Recall that|φn− f|^p→ 0 a.e. and |φn− f|^p≤ 2^p|f|^p∈ L¹, so by the Dominated Convergence Theorem,φn− fL^p → 0.

Now, suppose that the set E_i are disjoint; then b definition of the Lebesgue integral,



X

φ^p_ndx =

n i=1

|ai|^pμ(E_i) <∞ .

If ai= 0, then μ(Ei) <∞. 

1.7. Approximation of L^p(Ω)by continuous functions.

Lemma 1.24. Suppose that Ω ⊂ Rⁿ is bounded. Then C⁰(Ω) is dense in L^p(Ω) for p∈ [1, ∞).

Proof. Let K be any compact subset of Ω. The functions F_K,_n(x) = 1

1 + n dist(x, K) ∈ C⁰(Ω) satisfy F_K,_n≤ 1 ,

and decrease monotonically to the characteristic function1K. The Monotone Con- vergence Theorem gives

fK,n→ 1K in L^p(Ω), 1≤ p < ∞ .

Next, let A⊂ Ω be any measurable set, and let λ denote the Lebesgue measure.

Then

λ(A) = sup{μ(K) : K ⊂ A, K compact} .

It follows that there exists an increasing sequence of Kj of compact subsets of A such that λ(A\ ∪jK_j) = 0. By the Monotone Convergence Theorem,1Kj → 1Ain L^p(Ω) for p∈ [1, ∞). According to Lemma 1.23, each function in L^p(Ω) is a norm limit of simple functions, so the lemma is proved.  1.8. Approximation of L^p(Ω) by smooth functions. For Ω ⊂ Rⁿ open, for

 > 0 taken sufficiently small, define the open subset of Ω by Ω:={x ∈ Ω | dist(x, ∂Ω) > } . Definition 1.25 (Mollifiers). Define η ∈ C^∞(Rⁿ) by

η(x) :=

 Ce^{(|x|2−1)−1} if |x| < 1

0 if |x| ≥ 1 ,

with constant C > 0 chosen such that

Rⁿη(x)dx = 1.

For  > 0, the standard sequence of mollifiers on Rⁿ is defined by η(x) = ⁻ⁿη(x/) ,

and satisfy

Rⁿη(x)dx = 1 and spt(η)⊂ B(0, ).

NOTES ON L^p AND SOBOLEV SPACES 7

Definition 1.26. For Ω ⊂ Rⁿ open, set

L^p_loc(Ω) ={u : Ω → R | u ∈ L^p( ˜Ω) ∀ ˜Ω ⊂⊂ Ω} ,

where ˜Ω⊂⊂ Ω means that there exists K compact such that ˜Ω ⊂ K ⊂ Ω. We say that ˜Ω is compactly contained in Ω.

Definition 1.27 (Mollification of L¹). If f ∈ L¹_loc(Ω), define its mollification f^= η∗ f in Ω,

so that

f^(x) =



Ω

η_(x− y)f(y)dy =

B(0,)

η_(y)f (x− y)dy ∀x ∈ Ω. Theorem 1.28 (Mollification of L^p(Ω)).

(A) f^∈ C^∞(Ω).

(B) f^→ f a.e. as  → 0.

(C) If f ∈ C⁰(Ω), then f^→ f uniformly on compact subsets of Ω.

(D) If p∈ [1, ∞) and f ∈ L^p_loc(Ω), then f^→ f in L^p_loc(Ω).

Proof. Part (A). We rely on the difference quotient approximation of the partial derivative. Fix x ∈ Ω, and choose h sufficiently small so that x + hei ∈ Ω for i = 1, ..., n, and compute the difference quotient of f^:

f^(x + hei)− f(x)

 = ⁻ⁿ



Ω

1 h

 η

x + hei− y



− η

x− y h

f (y)dy

= ⁻ⁿ



Ω˜

1 h

 η

x + hei− y



− η

x− y



f (y)dy for some open set ˜Ω⊂⊂ Ω. On ˜Ω,

h→0lim 1 h

 η

x + hei− y



− η

x− y



= 1



∂η

∂xi

x− y



= ⁿ∂η

∂xi

x− y



, so by the Dominated Convergence Theorem,

∂f_

∂xi(x) =



Ω

∂η_

∂xi(x− y)f(y)dy .

A similar argument for higher-order partial derivatives proves (A).

Step 2. Part (B). By the Lebesgue differentiation theorem,

→0lim 1

|B(x, )|



B(x,)|f(y) − f(x)|dy for a.e. x ∈ Ω . Choose x∈ Ω for which this limit holds. Then

|f(x)− f(x)| ≤



B(x,)η(x− y)|f(y) − f(x)|dy

= 1

ⁿ



B(x,)

η((x− y)/)|f(y) − f(x)|dy

≤ C

|B(x, )|



B(x,)|f(x) − f(y)|dy −→ 0 as  → 0 .

8 STEVE SHKOLLER

Step 3. Part (C). For ˜Ω ⊂ Ω, the above inequality shows that if f ∈ C⁰(Ω) and hence uniformly continuous on ˜Ω, then f^(x)→ f(x) uniformly on ˜Ω.

Step 4. Part (D). For f ∈ L^p_loc(Ω), p∈ [1, ∞), choose open sets U ⊂⊂ D ⊂⊂ Ω;

then, for  > 0 small enough,

f^L^p(U)≤ fL^p(D). To see this, note that

|f^(x)| ≤



B(x,)

η_(x− y)|f(y)|dy

=



B(x,)η(x− y)^(p−1)/pη(x− y)^1/p|f(y)|dy

≤ 

B(x,)η(x− y)dy

(p−1)/p

B(x,)η(x− y)|f(y)|^pdy 1/p

, so that for  > 0 sufficiently small



U|f^(x)|^pdx≤



U



B(x,)η(x− y)|f(y)|^pdydx

≤



D|f(y)|^p 

B(y,)η(x− y)dx 

dy≤



D|f(y)|^pdy . Since C⁰(D) is dense in L^p(D), choose g ∈ C⁰(D) such thatf − gL^p(D) < δ;

thus

f^− fL^p(U)≤ f^− g^L^p(U)+g^− gL^p(U)+g − fL^p(U)

≤ 2f − gL^p(D)+g^− gL^p(U)≤ 2δ + g^− gL^p(U).

 1.9. Continuous linear functionals on L^p(X). Let L^p(X)^ denote the dual space of L^p(X). For φ ∈ L^p(X)^, the operator norm of φ is defined byφop = sup_Lp(X)=1|φ(f)|.

Theorem 1.29. Let p ∈ (1, ∞], q = _p−1^p . For g∈ L^q(X), define Fg: L^p(X)→ R as

F_g(f ) =



X

f gdx .

Then F_g is a continuous linear functional on L^p(X) with operator normFgop=

gL^p(X).

Proof. The linearity of Fgagain follows from the linearity of the Lebesgue integral.

Since

|Fg(f )| = 

Xf gdx ≤

X|fg| dx ≤ fL^pgL^q ,

with the last inequality following from H¨older’s inequality, we have that sup_f

Lp=1|Fg(f )| ≤

g_Lq.