Page 111 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes 
If E E {E : 1 i n}, n 1, 2, , then
n n i
E  n E n X 0 = F n F : 1 i n F n
i
= F F = 0,
n n
So that E  n E n = 0, and therefore



(F ) = (E ) (E ) E
n 1 0 n n 1 0 n n 1 0 n n 1 n
=  E  F
n 1 n 0 n 1 n
In other word is indeed a measure, and the proof of the theorem is complete.
0
9.2 Summary

 Let  be a -algebra of subsets of a set X. The pair (X, ) is called a measurable space. A
subset E of X is said to be -measurable if E .

 If is a measure on a -algebra of subsets of a set X, we call the triple (X, , ) a measure
space.
 A subset E of X is called a null set with respect to the measure if E  and (E) = 0.

 Two measurable spaces (X, ) and (Y, ). Let f be a mapping with D (f) X and  (f) Y.
–1
–1
We say that f is a /-measurable mapping if f (B)  for every B , that is f () .
9.3 Keywords

Complete Measure Space: Given a measure on a -algebra  of subsets of a set X. We say that
the -algebra  is complete with respect to the measure if an arbitrary subset E of a null set
0
E with respect to is a member of  (and consequently has (E ) = 0 by the Monotonicity of ).
0
When  is complete with respect to , we say that (X, , ) is a complete measure space.
Measurable Mapping: Given two measurable spaces (X, ) and (Y, ). Let f be a mapping with D
(f) X and  (f) Y. We say that f is a / measurable mapping if f (B) for every B , that
–1
–1
is, f () .
Measurable Space: A measurable space is a set S, together with a non-empty collection, S, of
subsets of S.
Null Set in a Measure Space: 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, , ).

Sigma Algebra:  is sigma algebra which establishes following relations:

(i) A  for all k implies  A 
k k
k 1
(ii) A implies A 
C
(iii) 

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