Kenneth Kuttler

12.4. CONTINUITY OF TRANSLATION 365

Proof: Let D̃ be the restrictions of D to Ω. If f ∈ Lp(Ω), let F be the zero extensionof f to all of Rn. Let ε > 0 be given. By Theorem 12.3.1 or 12.3.2 there exists s ∈D suchthat ∥F− s∥p < ε . Thus

∥s− f∥Lp(Ω,µ) ≤ ∥s−F∥Lp(Rn,µ) < ε

and so the countable set D̃ is dense in Lp(Ω). ■

12.4 Continuity of TranslationDefinition 12.4.1 Let f be a function defined on U ⊆Rn and letw ∈Rn. Then fwwill be the function defined on w+U by fw(x) = f (x−w).

Theorem 12.4.2 (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 <

3 . Youcan see this from looking at simple functions and passing to the limit or you could use thechange of variables formula to verify it.

Therefore

∥ f − fw∥p ≤ ∥ f −g∥p +∥g−gw∥p +∥gw− fw∥<2ε

3+∥g−gw∥p. (12.9)

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

1+mn (B)1/p) <

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

3 and so from 12.9 ∥ f − fw∥p < ε . ■Here is a remarkable corollary.

Corollary 12.4.3 Suppose f ∈ L1 (Rp,mp) and let v be any nonzero vector. Then thereis a set of measure zero N and a sequence tn→ 0+ such that if x /∈ N, and 0 < sn ≤ tn

limn→∞| f (x)− f (x+ snv)|= 0.

Proof: Let tn be such that if sn≤ tn, ∥ f − f (·+ snv)∥L1 < 4−n. This exists by continuityof translation in L1 (Rp,mp). Then

mp (En)≡ mp({x : | f (x)− f (x+snv)| ≥ 2−n})≤ ∫ | f − f (·+ snv)|dmp

2−n < 2−n

Thus there is a set of measure zero N such that if x /∈ N, then x is in only finitely many ofthe sets En. It follows that for all n sufficiently large | f (x)− f (x+snv)|< 2−n. ■

12.4. CONTINUITY OF TRANSLATION 365Proof: Let be the restrictions of J to Q. If f € L?(Q), let F be the zero extensionof f to all of R”. Let € > 0 be given. By Theorem 12.3.1 or 12.3.2 there exists s € Z suchthat ||F —s||,, < €. ThusIls —S lla) S Ils Flip cep) < €and so the countable set J is dense in L? (Q).12.4 Continuity of TranslationDefinition 12.4.1 Le f be a function defined on U C R" and let w € R". Then fwwill be the function defined on w +U by fw(x) = f(a —w).Theorem 12.4.2 (Continuity of translation in L? ) Let f € L?(IR") with the measurebeing Lebesgue measure. Then lim\\,,\-0 || fw — S| p = 0.Proof: Let € > 0 be given and let g € C.(R") with ||g—f||,, < 5. Since Lebesguemeasure is translation invariant (m,(w +E) = mn(E)), ||gw — fwl|p = |lg —fllp < §- Youcan see this from looking at simple functions and passing to the limit or you could use thechange of variables formula to verify it.Therefore2€lf — fwllp < lf —8llp + lls — 8wllp + 8 — fw] < 3 + ||8 —8wlp- (12.9)But lim),,|-,0 8w(#) = g(x) uniformly in x because g is uniformly continuous. Now let Bbe a large ball containing spt(g) and let 6; be small enough that B(a,6) C B wheneverx € spt(g). If € > 0 is given there exists 6 < 6; such that if |w| < 6, it follows that\g(a—w)—g(x)|<e/3 (Lm (B)'/”). Therefore,legally = (fie) —s@—wlfdm) " <e- To on aThus, whenever |w| < 6, it follows ||g — gw||p < 4 and so from 12.9 ||f — fw||p < €.Here is a remarkable corollary.Corollary 12.4.3 Suppose f < L' (R’ ,Mp) and let v be any nonzero vector. Then thereis a set of measure zero N and a sequence t, — 0+ such that if« ¢ N, and 0 < Sy < tylim |f (a) — f (w+ snv)| =0.n—coProof: Let, be such that if 5, < ty, || f —f (-+5nv)||;1 <4-". This exists by continuityof translation in L' (R’,m,). ThenJ\f=fC+5nv)|dmpQ-nMp (En) = Mp ({@: |f (aw) — f (w@tsnv)| >2°-"}) < <2”Thus there is a set of measure zero N such that if x ¢ N, then z is in only finitely many ofthe sets E,,. It follows that for all n sufficiently large | f (a) — f (a+s,v)|<27". I