26.3. WEAK DERIVATIVES 943

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

E ≡ { x : f (x)> ε}

and letEm ≡ E ∩B(0,m).

We show that 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, ≺, has the same meaning as it does in Chapter 12. Then let φ δ be amollifier 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).

Therefore, m(Em) = 0, a contradiction. Thus

m(E)≤∞

∑m=1

m(Em) = 0

and so, since ε > 0 is arbitrary,

m({ x : f ( x)> 0}) = 0.

Similarly m({ x : f ( x)< 0}) = 0. This proves the lemma.Example: δ x ∈D∗ (Ω) where δ x (φ)≡ φ (x).It will be observed from the above two examples and a little thought that D∗ (Ω) is

truly enormous. We shall define the derivative of a distribution in such a way that it agreeswith the usual notion of a derivative on those distributions which are also continuouslydifferentiable functions. With this in mind, let f be the restriction to Ω of a smooth functiondefined on Rn. Then Dxi f makes sense and for φ ∈C∞

c (Ω)

Dxi f (φ)≡∫

Ω

Dxi f (x)φ (x)dx =−∫

Ω

f Dxiφdx =− f (Dxiφ).

Motivated by this, here is the definition of a weak derivative.