Page 321 - DMTH401_REAL ANALYSIS

Unit 26: Lebesgue Measure

Theorem 8: Littlewood’s 3rd Principle/Egoroff’s Theorem Notes
Let E  M with m(E) < ¥, f: E  (–¥, ¥) be measurable and { } n be a sequence of measurable
f
n
functions on E such that
f  f a.e. on E.
n
Then for any  > 0 there is a (measurable) subset S of E with m(S) <  such that
f  f uniformly on E\S.
n

Notes Again, the condition m(E) < ¥ cannot be dropped. Otherwise f =  and f = 0
n [n, ¥)
would be a counter example.
Proof: We claim that for any  > 0 and  > 0, there exists A  E with m(A) <  and N   such that
|f (x) – f(x)| <  whenever n  N and x  E \A.
n
Be careful the above statement is not saying that f  f uniformly on E \A since A depends on
n
 and .
To prove our claim, we let
G = {x  E : |f (x) – f(x)|  }
n n
and
G = lim G :=  E , where E =  G .
n n n k
n k n

Note that if x  G then x  E for all n  , it follows that f (x)  f(x). Since the set of all x such
n n
that f (x)  f(x) is of measure zero, we have m(G) = 0. Note also that m(E ) < ¥ and E “decreases”
n 1 n
to G, so limm(E ) = m(G) = 0 by Lemma 4. There is N   such that m(E ) < . This N, together
n N
with A : = E , proved our claim.
N
k
Now, let  > 0 be given. Apply the above result to = 1/k and  = /2 , we obtain A with m(A )
k k
k
< /2 and N   such that
k
1
|f (x) – f(x) | < whenever n  N and x  E \A .
n k k
k
Let S =  k A , then m(S)  å ¥ k 1 m(A ) <  and |f (x) – f(x)| < 1/k whenever n  N and x  E\S.
=
k
n
k
k
Hence, f  f uniformly onE\S.
n
Self Assessment
Fill in the blanks:
1. Every open subset V of  is a ........................... of disjoint open intervals.
2. The family of all measurable sets is denoted by M. We will see later M is a -algebra and
translation-invariant containing all intervals. The set function m: M  [0, ¥] defined by
m(E) = m*(E) for all E  M
is called .....................................
3. Let X be a .............................. and Y be a topological space. A function f: X  Y is called
measurable if f (V) is a measurable set in X for every open set V inY.
–1
4. Let E  M, f: E  [–¥, ¥] and g: E  [–¥, ¥]. If f = g almost everywhere on E then the
................................ of f and g are the same.

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