Page 145 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes 11.2 Summary

 Let and be simple functions which vanish outside a set of finite measure, then

(a b ) a b and if a.e., then

 A Lebesgue integrable function may not be Riemann integrable.
 Let A be the set of all irrational numbers and A be the set of all rational numbers in [0, 1].
1 2
 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} <

 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

11.3 Keywords

Canonical Representation: If is simple function and { , , …, } the set of non-zero values of
1 2 n
, then
n
= ,
i E i
i 1
where E = {x : (x) = }.
i i
Characteristic Function: The function defined by
E
1 if x E
(x) =
E 0 if x E

is called the characteristic function of E.
Elementary Integral: If vanishes outside a set of finite measure, we define the elementary
n
integral of by (x) dx i mE when has the canonical representation.
i
i 1
n
= i E i .
i 1
Lebesgue Integrable: A measurable function f is said to be Lebesgue integrable over E if f and f –
+
are both Lebesgue integrable over E. In this case, we define f f f .
E E E

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