240 CHAPTER 10. REGULAR MEASURES
≤ mp
([x : limsup
r→0
1mp (B(x,r))
∫B(x,r)
| f (y)−g(y)|dmp (y)>ε
2
])+mp
([x : limsup
r→0
1mp (B(x,r))
∫B(x,r)
|g(y)−g(x)|dmp (y)>ε
2
])+mp
([x : |g(x)− f (x)|> ε
2
]).
≤ mp
([M ( f −g)>
ε
2
])+mp
([| f −g|> ε
2
])(10.3)
Now∥ f −g∥1 ≥
∫[| f−g|> ε
2 ]| f −g|dmp ≥
ε
2mp
([| f −g|> ε
2
])and so using Claim 1 and 10.3, 10.2 is dominated by
(2ε+ 5p
ε
)∫| f −g|dmp. But by Propo-
sition 10.1.4, g can be chosen to make the above as small as desired. Hence 10.2 is 0.
mp
([limsup
r→0
1mp (B(x,r))
∫B(x,r)
| f (y)− f (x)|dmp (y)> 0])
≤∞
∑k=1
mp
([limsup
r→0
1mp (B(x,r))
∫B(x,r)
| f (y)− f (x)|dmp (y)>1k
])= 0
By completeness of mp this implies[limsup
r→0
1mp (B(x,r))
∫B(x,r)
| f (y)− f (x)|dmp (y)> 0]
is a set of mp measure zero. ■The following corollary is the main result referred to as the Lebesgue Differentiation
theorem.
Definition 10.2.2 f ∈ L1loc (Rp,mp) means f XB is in L1 (Rn,mp) whenever B is a
ball.
Corollary 10.2.3 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 . (10.4)
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 10.4 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, · · ·}, 10.4 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)