1236 CHAPTER 35. WEAK DERIVATIVES

Also define φ k to be a mollifier if

spt(φ k)⊆ B(

0,1k

),φ k ≥ 0

,∫

φ kdx = 1, and φ k ∈ C∞c(B(0, 1

k

)). In the case a Greek letter like δ or ε is used as a

subscript, it will mean spt(φ δ )⊆ B(0,δ ) ,φ δ ≥ 0,∫

φ δ dx = 1, and φ δ ∈C∞c (B(0,δ )) . You

can always get a mollifier by letting φ ≥ 0,φ ∈C∞c (B(0,1)) ,

∫φdx = 1,and then defining

φ k (x)≡ knφ (kx) or in the case of a Greek subscript, φ δ (x) =1

δn φ( x

δ

).

Consider the case where u and Dα u for |α|= 1 are each in Lploc (R

n). The next lemmais the one alluded to in the proof of Theorem 35.2.2.

Lemma 35.3.3 Suppose f ∈ L1loc (U) and suppose∫

f φdx = 0

for all φ ∈C∞c (U). Then f (x) = 0 a.e. x.

Proof: Without loss of generality f is real valued. Let

E ≡ {x : f (x)> ε}

and letEm ≡ E ∩B(0,m).

Is m(Em) = 0? If not, there exists an open set, V , and a compact set K satisfying

K ⊆ Em ⊆V ⊆ B(0,m) , m(V \K)< 4−1m(Em) ,∫V\K| f |dx < ε4−1m(Em) .

Let H and W be open sets satisfying

K ⊆ H ⊆ H ⊆W ⊆W ⊆V

and letH ≺ g≺W

where the symbol, ≺, in the above implies spt(g) ⊆ W, g has all values in [0,1] , andg(x) = 1 on H. Then let φ δ be a mollifier and let h≡ g∗φ δ for δ small enough that

K ≺ h≺V.

Thus

0 =∫

f hdx =∫

Kf dx+

∫V\K

f hdx

≥ εm(K)− ε4−1m(Em)

≥ ε(m(Em)−4−1m(Em)

)− ε4−1m(Em)

≥ 2−1εm(Em).