Page 140 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 11: Integration

Notes
1
f dx > mE
n n
E
1
Or mE n < f dx
n
E
But f dx < 0
E

1
mE < 0
n n
m E < 0
n
But m E 0 is always true
n
m E = 0
n

But {x : f (x) > 0} =  E n
n 1
and m E = 0
n
m  E = 0
n
n 1
m {x : f (x) > 0} = 0
f = 0 a.e. on E

Theorem 5: Let f and g be two non-negative measurable functions. If f is integrable over E and
g (x) < f (x) on E, then g is also integrable over E, and

(f g) = f g .
E E E

Proof: Since 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
Since f = (f – g) + g,
therefore we have

f = (f g g) (f g) g … (1)
E E E E
Since the functions f – g and g are non-negative and measurable. Further, f being integrable over
E, f < (by definition)
E

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