Page 143 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes Furthermore f and g are integrable, it implies that |f| and |g| are integrable.
( A measurable function f is integrable over E if and only if |f| is integrable over E.)
Thus |f| + |g| is integrable over E.

( (f g) f g and by the definition of integrable)
E E E
Since |f + g| |+|f| + |g|

which shows that f + g is integrable.
Hence sum of two integrable functions is integrable.

Thus (f g) = (f g ) (f g )
E E E

= f g f g
E E E E

= f f g g
E E E E

= f g
E E
(c) f g a.e.
f – g 0 a.e.
g – f 0 a.e.

(g f) 0
E

Since g = f + (g – f) and f, g – f are integrable over E.
–
Then by the given hypothesis (g – f) = 0 a.e.
then (g f) = 0,
E

(Since we know that if f = 0 a.e. then f = 0)
E

g f (g f) f (g f) (g f)
E E E E E E
becomes

g = f (g f) 0 f (g f) ( (g – f) 0)
E E E E E

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