Page 175 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes
Proof: Let P =  {F a F }
b .
b a
Evidently (P) = 0.

Let F a = F P.
a
This F F b = (F – F ) – P = for a < b.
a a b
In view of Lemma 1, it follows that a measurable function f s.t.
f (x) a, x F
a
and f (x) a, x X – F
a
Thus we have
x F a f(x) a a.e., except for x P.
x X F f(x) a a.e.,
a

Proof of the main theorem

At first, suppose that is finite.
( – a ) is a signed measure on  for each rational number a.
Let (P , Q ) be a Hahn decomposition for the measure ( – a ).
a a
Let P = X and Q = .
O O
By the definition of Hahn decomposition theorem,
P Q = X,
a a
and P Q = X.
b b
Therefore, Q – Q = Q – (X – P )
a b a b
= Q P .
a b
Thus, ( – a ) (Q – Q ) 0 … (i)
a b
Similarly, we can prove that

( – b ) (Q – Q ) 0 … (ii)
a b
Let a < b, then from (i) and (ii), we have
(Q – Q ) = 0.
a b
Therefore, by Lemma (ii)
f (x) a, a.e. x P
a
and f (x) a, a.e. x Q ,
a
where f is measurable
Since Q = , it follows that f is non-negative
O
Again, let A  be arbitrary.
Q Q
Define A = A r 1 r
r
n n
o o

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