Page 138 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 11: Integration

Notes
f g
E E
Since f |f|

f |f| … (1)
E E
Again – f |f|

f |f|
E E

or |f| f … (2)
E E
From (1) and (2) we get

|f| f |f|
E E E

f |f|
E E

Proof of 4: It follows from (3) and the fact that 1 mE .
E

Proof of 5: f = f
A B
A B
Now A B = A B A B

where A and B are disjoint measurable sets i.e.
A B =

f = f( A B ) f A B
A B

= f f 0 [ A B = and m ( ) = 0]
A B

= f f
A B

11.1.3 The Lebesgue Integral of a Non-negative Function

Definition: If f is a non-negative measurable function defined on a measurable set E, we define

f = sup h ,
h f
E E

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