Page 109 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes By the countable subadditivity of on ,
we have (E) (E ) = 0.
n  n
Thus (E) = 0.
This shows that E is a null set in (X, , ).

9.1.2 Complete Measure Space

Definition: 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 E with respect to
0
is a member of  (and consequently has  (E ) = 0 by the Monotonicity of ). When  is
0
complete with respect to , we say that (X, , ) is a complete measure space.

Example: Let X = {a, b, c}. Then = { , {a}, {b, c}, X} is a -algebra of subsets of X. If we
define a set function on by setting ( ) = 0, ({a}) = 1, ({b, c}) = 0, and (X) = 1, then is
a measure on . The set {b, c} is a null set in the measure space (X, , ), but its subset {b} is not
a member of . Therefore, (X, , ) is not a complete measure space.

9.1.3 Measurable Mapping

Let f be a mapping of a subset D of a set X into a set Y. We write D (f) and R (f) for the domain of
definition and the range of f respectively. Thus
D (f) = D X,

R (f) = {y Y : y = f (x) for some x D (f)} Y.
For the image of D (f) by f, we have f (D (f)) = R (f). For an arbitrary subset E of y we define the
preimage of E under the mapping f by

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F (E) : = {x X : f (x) E} = {x D (f) : f (x) E}.

Notes
1. E is an arbitrary subset of Y and need not be a subset of R (f). Indeed E may be disjoint
from  (f), in which case f (E) = . In general, we have f (f (E)) E.
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2. For an arbitrary collection C of subsets of Y, we let f (C) : = {f (E) : E C}.
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Theorem 2: Given sets X and Y. Let f be a mapping with D (f) X and  (f) Y. Let E and E be
arbitrary subsets of Y. Then
1. f (Y) = D (f),
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2. f (E ) = f (Y\E) = f (Y)\f (E) = D (f) \ f (E),
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C
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3. f (U E ) = U f (E ),
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 
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4. f ( E ) = f (E ).
 
Theorem 3: Given sets X and Y. Let f be a mapping with D (f) X and R (f) Y. If  is a -algebra
of subsets of Y then f () is a -algebra of subsets of the set D (f). In particular, if D (f) = X then
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f () is a -algebra of subsets of the set X.
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