Page 371 - DMTH401_REAL ANALYSIS
P. 371
Unit 30: Riemann's and Lebesgue
for every n and hence ¥ n 1 A +1 . Thus is a -algebra on containing all open Notes
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=
n
subsets of . Hence = ( ).
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Now, it suffices to show that card() = card(). Since there is an open ball of radius 1 centered at
each point of , we have card() card( ) card(). So it suffices to establish that card()
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card(). Since card(L ) = card() and = < , it is enough to show that card( ) card() for
each L .
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Let be the collection of all open balls in with rational radius and center in . Then is
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countable, and any open set U can be written as a countable union of members of . Hence
card( ) card( ) = card(). Let L and suppose we have proved that card( ) card() for
every < . If is a limit ordinal, then = < is a countable union and hence card( )
card(). If there is with + 1 = , then any A can be written as A = ¥ A with A A
=
n 1 n n
. This gives a one-one map from into (A ) . Hence card( ) card((A ) )
card(). This completes the proof.
Corollary: For any uncountable set Y , there is A Y such that A is not a Borel subset of .
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Proof: We have card(( )) = card() = card(Y) < card((Y)).
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Definition: Let ( ) = {A : m * L,d [A] = 0}. The members of ( ) are called Lebesgue null sets.
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The -algebra ( ) on generated by ( ) ( ) is called the Lebesgue -algebra on ,
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and members of ( ) are called Lebesgue measurable subsets of .
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card(( )) = card(( )) = card(( )) > card(). Hence, ( ) ( ) ( ).
Proof: Let K be the middle-third Cantor set. Then, for any subset A K, we have m * L,1 [A]
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m * [K] = 0. So m * [A] = 0 also. This shows that (K) ( ) ( ). And card((K)) = card(())
L,1 L,d
since K is an uncountable subset of .
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[Translation invariance] (i) A + x ( ) for every A ( ) and x .
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(ii) A + x ( ) for every A ( ) and x .
(iii) A + x ( ) for every A ( ) and x .
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Proof: First let us mention a general principle that will be used at many places. To establish that
the members of a certain -algebra on a set X satisfies a certain property P, it suffices to do
the following: show that the collection {A X: A satisfies property P} is a -algebra, and then
find a suitable collection (X) generating and show that every member of satisfies the
property P.
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Let = {A : A + x ( ) for every x }. It is easy to check that is a -algebra containing
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all d-boxes. And recall that the collection of all d-boxes generates ( ). This proves (i). Next,
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statement (ii) is a consequence of the translation invariance property of the Lebesgue outer
measure, and (iii) follows from (i) and (ii) by applying the principle mentioned above.
We will give other characterizations of the Lebesgue measurable sets shortly, and we will also
show that ( ) ( ).
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Self Assessment
Fill in the blanks:
1. ............................................... is developed through approximations of a finite nature (e.g.:
one tries to approximate the area of a bounded subset of by the sum of the areas of
2
finitely many rectangles).
2. While Riemann’s theory is restricted to the Euclidean space, the ideas involved in
............................ are applicable to more general spaces, yielding an abstract measure theory.
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