9.6. A USEFUL INEQUALITY 233

Definition 9.5.2 f ∈ L1loc (Rp,mp) means f XB is in L1 (Rn,mp) whenever B is a

ball.

Corollary 9.5.3 If f ∈ L1loc (Rp,mp), then for a.e.x,

limr→0

1mp (B(x,r))

∫B(x,r)

| f (y)− f (x)|dmp (y) = 0 . (9.10)

In particular, for a.e.x,

limr→0

1mp (B(x,r))

∫B(x,r)

f (y)dmp (y) = f (x)

Proof: If f is replaced by f XB(0,k) then the conclusion 9.10 holds for all x /∈ Fk whereFk is a set of mp measure 0. Letting k = 1,2, · · · , and F ≡ ∪∞

k=1Fk, it follows that F is aset of measure zero and for any x /∈ F , and k ∈ {1,2, · · ·}, 9.10 holds if f is replaced byf XB(0,k). Picking any such x, and letting k > |x|+1, this shows

limr→0

1mp (B(x,r))

∫B(x,r)

| f (y)− f (x)|dmp (y)

= limr→0

1mp (B(x,r))

∫B(x,r)

∣∣ f XB(0,k) (y)− f XB(0,k) (x)∣∣dmp (y) = 0.

The last claim holds because∣∣∣∣ f (x)− 1mp (B(x,r))

∫B(x,r)

f (y)dmp (y)∣∣∣∣≤ 1

mp (B(x,r))

∫B(x,r)

| f (y)− f (x)|dmp (y) ■

Definition 9.5.4 Let E be a measurable set. Then x ∈ E is called a point of densityif

limr→0

mp (B(x,r)∩E)mp (B(x,r))

= 1

Proposition 9.5.5 Let E be a measurable set. Then mp a.e. x ∈ E is a point of density.

Proof: This follows from letting f (x) = XE (x) in Corollary 9.5.3. ■

9.6 A Useful InequalityThere is an extremely useful inequality. To prove this theorem first consider a special caseof it in which technical considerations which shed no light on the proof are excluded.

Lemma 9.6.1 Let (X ,S ,µ) and (Y,F ,λ ) be finite measure spaces and let f be µ×λ

measurable and uniformly bounded. Then the following inequality is valid for p≥ 1.

∫X

(∫Y| f (x,y)|p dλ

) 1p

dµ ≥(∫

Y(∫

X| f (x,y)|dµ)pdλ

) 1p

. (9.11)