Page 350 - DMTH401_REAL ANALYSIS
P. 350
Real Analysis
Notes But the A Ç B’s are disjoint. So by Lemma we have
i j
n m
A ò = å å a m(A Ç B Ç A)
i
i
j
=
=
i 1 j 1
n m
A ò = å å b m(A Ç B Ç A)
j
i
i
i 1 j 1
=
=
and
n m
)
A ò ( + = å å (a + b )m(A Ç B Ç A).
j
i
i
j
=
=
i 1 j 1
)
Hence ò ( + = A ò +
A
n
(b) If a = 0 the result is trivial; if not, then let = å i 1 a A i be the canonical representation of
=
i
n
. We see that a = å i 1 aa A is the canonical representation of a and hence the result
=
i
i
follows.
)
(c) Since ò - A ò = A ò ( - by linearity, it suffices to show ò f 0 whenever 0 a.e. on
A A
A. This is easy, since if a , a , . . ., a are the distinct values of f, then
1 2 n
A ò f å a m(f - 1 {a } Ç A) + å a m( f - 1 {a } Ç A) å a 0 = 0
i
i
i
i
i
{i:a i 0}< {i:a i 0} {i:a i 0}<
–1
where the inequality follows from the fact that m(f{a Ç A}) = 0 for all a < 0.
i i
(d) This is immediate from (c).
(e) Since it is given that 0 a.e. on A, it suffices to show m({x : (x) > 0} Ç A) = 0.
Suppose not, then there exists a > 0 such that m({x : (x) = a} Ç A) > 0 so ò a m ({x : (x)
A
= a}Ç A) > 0. This leads to a contradiction.
(f) This follows directly from monotonicity since –|| ||.
29.3 Bounded Measurable Functions Vanishing Outside a
Set of Finite Measure
Resembling the construction of the Riemann integral, we define the upper and lower Lebesgue
integrals.
Definition: Let f : E be a bounded function which vanish outside a set of finite measure. For
any A C, we define the upper integral and the lower integral of f on A by
___
ò f inf {ò A :f on A, Î So(E) }
=
A
ì ü
ò f supí ò : f on A, Î So(E)ý
=
____ A î A þ
If the two values agree we denote the common value by ò A f . (Again the set A is implicitly
assumed to be measurable so that A ò and ò make sense.)
A
Note that both the infimum and the supremum in the definitions of the upper and lower integrals
exist because f is bounded and vanishes outside a set of finite measure. It is evident that for the
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