Page 108 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 9: Measure Spaces

(c) A measure on a -algebra  of subsets of a set X is called a -finite measure if there exists Notes
a sequence (E : n ) in  such that  E = X and (E ) < for every n . In this case
n n  n n
(X, , ) is called a -finite measure space.
(d) A set D  in an arbitrary measure space (X, , ) is called a -finite set if there exists a
sequence (D : n ) in such that U D = D and (D ) < for every n .
n n  n n
Lemma 1: (a) Let (X, , ) be a measure space. If D  is a -finite set, then there exists an
increasing sequence (F : n ) in  such that limF D and (F ) < for every n  and there
n n n
n
exists a disjoint sequence (G : n ) in such that  G = D and (G ) < for every n .
n n  n n
(b) If (X, , ) is a -finite measure space then every D  is a -finite set.
Proof 1: Let (X, , ) be a measure space. Suppose D  is a -finite set. Then there exists a
sequence (D : n ) in  such that U D = D and (D ) < for every n . For each n ,
n n  n n
let F U n D . Then (F : n ) is an increasing sequence in  such that limF U F
n k 1 k n n n  n
n
n
U n  D n D and (F )  D k n (D ) for every n .
n
k
k 1 k 1
Let G = F and G = F \ U n 1 F for n 2. Then (G : n ) is a disjoint sequence in  such that
1 1 n n k 1 k n
U G U F D as in the proof of Lemma “let (E : n ) be an arbitrary sequence in an
n  n n  n n
algebra of subsets of a set X. Then there exists a disjoint sequence (F : n ) in such that
n
N N
 n 
(1) E F for every N ,
n
n 1 n 1
and
 n 
(2) E F ”.
n
n  n 

(G ) = (F ) < and (G ) F  n 1 F (F ) for n 2. This proves (a).
1 1 n n k 1 k n

2. Let (X, , ) be a -finite measure space. Then there exists a sequence (E : n ) in  such
n
that U E = X and (E ) < for every n . Let D . For each n , let D = D E .
n  n n n n
Then (D : n ) is a sequence in such that U D = D and m (D ) (E ) < for every
n n  n n n
n . Thus D is a -finite set. This proves (b).
9.1.1 Null Set in a Measure Space

Definition: Given a measure on a -algebra  of subsets of a set X . A subset E of X is called a null
set with respect to the measure if E  and (E) = 0. In this case we say also that E is a null set
in the measure space (X, , ). (Note that is a null set in any measure space but a null set in a
measure space need not be .)
Theorem 1: A countable union of null sets in a measure space is a null set of the measure space.
Proof: Let (E : n ) be a sequence of null sets in a measure space (X, , ). Let E = U E . Since
n n  n
 is closed under countable unions,
we have E .

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