Page 146 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 11: Integration

Notes
n
Simple Function: A linear combination (x) = i E i (x) is called a simple function if the sets
i 1
E are measurable.
i
n
Simple Function: A linear combination (x) = i E i (x) is called a simple function if the sets E i
i 1
are measurable.
This representation of is not unique.
However, a function is simple if and only if it is measurable and assumes only a finite number
of values.
The Lebesgue Integral of a Non-negative Function: If f is a non-negative measurable function
defined on a measurable set E, we define

f = sup h ,
h f
E E
where h is a bounded measurable function such that
m {x : h (x) 0} < 
The Riemann Integral: Let f be a bounded real valued function defined on the interval [a, b] and
let a = x < x < … < x = b be a sub-division of [a, b].
0 1 n
Then for each sub-division we can define the sums

n
S = (x x ) M
i i 1 i
i 1
n
and s = (x x ) m
i i 1 i
i 1
where M = sup f(x) ,
i
x i 1 x x i
m = inf f(x)
i
x i 1 x x i
11.4 Review Questions

1. Prove that af a f real number a.
E E
2. If f is bounded real valued measurable function defined on a measurable set E of finite

measure such that a f (x) b, then show that amE f bmE.
E
3. If f and g are non-negative measurable functions defined on E  then prove that

(a) cf c f, c 0
E E

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