Page 322 - DMTH401_REAL ANALYSIS

Real Analysis

Notes 26.7 Summary

 The definition of outer measure of sets.
 Outer measure of an interval is its length.
 Some important properties of Outer measure.

 The definition of Measurable sets.
 Countable union of measurable sets is also measurable.
 Countable intersection of measurable sets is also measurable.
 Every Borel set is measurable.
 Littlewood’s First Principle.

26.8 Keywords

Lindelof’s Theorem: Let C be a collection of open subsets of . Then there is a countable
sub-collection {O } i i of C such that
¥
 O =  O
O C i 1 i
=

Lebesgue Measure: 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 Lebesgue measure.

¥
 O =  O i
=
O C i 1

Littlewood’s 1st Principle: Every measurable set of finite measure is nearly a finite union of
disjoint open intervals, in the sense.
Measurable Functions: A function f: E  [-¥, ¥] is said to be measurable (or measurable on E) if
E  M and

f ((a, ¥])  M
–1
for all a  .

26.9 Review Questions

1. Prove that the family M of measurable sets is an algebra.
2. If E , E , ….En are measurable, prove that E U E …  E is measurable.
1 2 1 2 n
3. If E and E are measurable sets, then prove that E  E is also measurable.
1 2 1 2
4. Prove that properties (i) to (v) are equivalent to (vi), if m*E is finite.
5. Show that if E is measurable, then each translate E + y is also measurable.

6. Show that if E and E are measurable, then m(E  E ) + m(E |E ) = mE + mE .
1 2 1 2 1 2 1 2
7. Let {E } be a sequence of disjoint measurable sets and A be any set.
i
)
(
¥
Show that m* A   ¥ E = å m *(A  E )
i
i
i 1 i 1
=
=
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