8 I. Theory of Distributions

Taking real-valued ϕ ∈ D and separating the real and the imaginary part of

the integral, we can assume without loss of generality that f1(x) − f2(x) is

real-valued. Let

sgn(f1(x) − f2(x)) =

⎧

⎨

⎩

1 if f1(x) − f2(x) 0,

0 if f1(x) − f2(x) = 0,

−1 if f1(x) − f2(x) 0.

Then sgn(f1 − f2) is a bounded measurable function.

Fix an arbitrary R 0. It is known from Lebesgue measure theory (see,

for example, [R]) that there exists a sequence of step-functions {sm(x)}

such that sm(x) → sgn(f1 − f2) almost everywhere in BR as m → ∞ and

|sm(x)| ≤ 1. For any step-function sm(x), there exists a sequence of continu-

ous functions {cp(x)} with supports in BR such that cp(x) → sm(x) almost

everywhere (a.e.) in BR as p → ∞ and |cp(x)| ≤ 1. Using these two facts

and the Proposition 1.3 we can find a sequence ϕm(x) of C0

∞-functions

with

supports in BR such that ϕm(x) → sgn(f1 −f2) a.e. in BR and |ϕm(x)| ≤ 1.

Then the Lebesgue convergence theorem implies that

(f1(x) − f2(x))ϕm(x)dx →

BR

|f1(x) − f2(x)|dx as m → ∞.

Thus

BR

|f1(x) − f2(x)|dx = 0. Therefore f1(x) = f2(x) a.e. in BR. Since

R is arbitrary, we get f1(x) = f2(x) a.e. in

Rn.

The distributions form a linear space since a linear combination α1f1 +

α2f2 of distributions is also a distribution:

(α1f1 + α2f2)(ϕ) = α1f1(ϕ) + α2f2(ϕ).

The linear space of distributions is denoted by D .

2.3. Distributions in a domain.

Let Ω ⊂

Rn

be an arbitrary domain. The space D(Ω) is the space C0

∞(Ω)

with the following notion of convergence: ϕn(x) ∈ C0

∞(Ω)

converges to

ϕ(x) ∈ C0

∞(Ω)

in D(Ω) if there exists a compact subdomain B ⊂ Ω such

that supp ϕn ⊂ B for all n ≥ 1 and

∂kϕn(x)

∂xk

→

∂kϕ

∂xk

uniformly on B for all

0 ≤ |k| +∞.

The space of distribution D (Ω) is the space of all linear continuous

functionals on D(Ω).

Let U ⊂

Rn

be an open set and let f ∈ D

(Rn).

We shall define the

restriction f|U of f to U as f|U (ϕ) = f(ϕ) for all ϕ ∈ D(U). It is obvious

that f|U ∈ D (U).

Example 2.1 (A nonextendible distribution). Let f(x1) =

ex1

2

. Consider

the regular functional in D (0, +∞) defined by f(ϕ) =

∞

0

e

1

x2

1

ϕ(x1)dx1,