Page 384 - DMTH401_REAL ANALYSIS

Real Analysis

Notes = f + g – f – g –
–
+
+
+
–
= (f – f ) + (g – g )
+
–
= f + g.
Hence (ii) is proved.
–
–
(iii) Since f g a.e., f - f g – g a.e.,
+
+
–
+
+
Hence, f + g – g + f a.e,
+
(f + g ) (g + f ).
–
–
+
Hence
f + g – g + f .
+
–
+
Hence,
f – f – g – g –
+
+
Hence,
f g.
Hence (iii) is proved.
(iv) Consider
ò A B f = f ò ×c A B
È
È
= ò f (c + c B )
×
A
= f ò ×c + f ò ×c
A B
= A ò f + B ò f
32.2 Lebesgue Convergence Theorem
Theorem 2: Let g be integrable over E and let {f } be a sequence of measurable functions such that
n
|f |g on E and for almost all x in E we have f(x) = lim fn(x). Then
n
E ò f = lim f n
E ò
Proof: Since |f |g on E, g – f is nonnegative and hence by Fatou’s Lemma,
n n
-
-
E ò (g f)lim (g f ) ...(1)
E ò
n
Since f(x) = lim f (x) a.e. on E and
n
|f |g on E,
n
|f|g on E.
Hence since g is integrable,
f is also integrable.
-
E ò (g f) = E ò g - E ò f ...(2)
Also,
lim (g f ) E ò = g lim f n ...(3)
-
-
E ò
E ò
n
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