Page 142 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 11: Integration

Proof: (a) If c 0, then Notes
+
(cf) = cf +
–
(cf) = cf
and if c < 0, then
.
*
(cf) = (–c) f
.
(cf) = (–c) f +
–
–
+
Since f is integrable so f and f are also integrable and conversely. Hence the result
follows.
(b) In order to prove the required result first of all we show that if f and f are non-negative
1 2
integrable functions such that f = f – f , then
1 2
f = f f … (1)
1 2
E E E
Since f = f – f , –
+
Also f = f – f ,
1 2
+
–
then f – f = f – f
1 2
f + f = f + f – … (2)
+
2 1
Also we know that if f and g are non-negative measurable functions defined on a set E,
then
(f g) = f g
E E E
Then from (2), we get

f f = f f
2 1
E E E E

f f = f 1 f 2 … (3)
E E E E
–
+
But f is integrable so f and f are integrable i.e.
f = f f

Therefore (3) becomes

Hence f = f f
1 2
E E E
which proves (1).
Now, if f and g are integrable functions over E, then

–
+
–
–
–
+
f + g , f + g and f + g = (f + g ) – (f + g )
+
+
and also integrable functions over E.
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