10.6. CONTINUITY OF TRANSLATION 255

Theorem 10.5.8 Let U be an open set in Rn and let f ∈ Lp (U). Then there existsh ∈C∞

c (U) such that (∫U| f −h|p dmn

)1/p

< ε.

In words, C∞c (U) is dense in Lp (U).

Functions which vanish off a compact set are said to have “compact support”. Note thatall of this would work for any regular measure µ . Now what follows will be dependent onthe measure being Lebesgue measure or something like it.

10.6 Continuity of TranslationThis is a property which directly exploits density of continuous functions with compactsupport and the translation invariance of Lebesgue measure.

Definition 10.6.1 Let f be a function defined on U ⊆ Rn and let w ∈ Rn. Then fwwill be the function defined on w+U by

fw(x) = f (x−w).

We will write spt(g) to indicate the closure of the set on which g is nonzero. This is calledthe support of the function.

Theorem 10.6.2 (Continuity of translation in Lp) Let f ∈ Lp(Rn) with the measurebeing Lebesgue measure. Then

lim∥w∥→0

∥ fw− f∥p = 0.

Proof: Let ε > 0 be given and let g ∈ Cc(Rn) with ∥g− f∥p < ε

3 . Since Lebesguemeasure is translation invariant (mn(w+E) = mn(E)),

∥gw− fw∥p = ∥g− f∥p <ε

3.

You can see this from looking at simple functions and passing to the limit or you could usethe change of variables formula to verify it.

Therefore

∥ f − fw∥p ≤ ∥ f −g∥p +∥g−gw∥p +∥gw− fw∥

<2ε

3+∥g−gw∥p. (10.11)

But lim|w|→0 gw(x) = g(x) uniformly in x because g is uniformly continuous. Now let Bbe a large ball containing spt(g) and let δ 1 be small enough that B(x,δ ) ⊆ B wheneverx ∈ spt(g). If ε > 0 is given there exists δ < δ 1 such that if |w| < δ , it follows that|g(x−w)−g(x)|< ε/3

(1+mn (B)

1/p)

. Therefore,

∥g−gw∥p =

(∫B|g(x)−g(x−w)|p dmn

)1/p

≤ εmn (B)

1/p

3(

1+mn (B)1/p) <

ε

3.

Therefore, whenever |w| < δ , it follows ∥g−gw∥p <ε

3 and so from 10.11 ∥ f − fw∥p < ε .■