Page 134 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 11: Integration

Notes
Thus f dx f dx

The function is not Riemann integrable.

Now for Lebesgue Integrability
Let A be the set of all irrational numbers and A be the set of all rational numbers in [0, 1].
1 2
The partition P = {A , A } is a measurable partition of [0, 1] and mA = 0, mA = 1.
1 2 1 2
L (p, f) = inf f(x) mA 1 inf f(x) mA 2
A 1 A 2

= 0 mA 1 mA = 1.
1 2
U (p, f) = sup f(x) mA 1 sup f(x) mA 2
A 1 A 2

= 0 mA 1 mA = 1.
1 2
sup L (p, f) = 1 inf U(p, f)
p p
f is Lebesgue integrable over [0, 1].
Theorem 3: If f and g are bounded measurable functions defined on the set E of finite measure,
then

(1) (af bg) a f b g
E E E

(2) If f = g a.e., then f g
E E

(3) If f g a.e., then f g
E E

Hence f |f|

(4) If A and B are disjoint measurable set of finite measure, then

f f f
A B A B
Proof of 1: Result is true if a = 0
Let a 0.

If is a simple function then so is a and conversely.
Hence for a > 0

af = inf a
a af
E E

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