Page 380 - DMTH401_REAL ANALYSIS

Real Analysis

Notes
7. If {f }, {g } are sequences of measurable functions defined on E, |f | £ g , f = lim f , g =
n n n n n n
liminf g and ...................................., then lim ò f exists and is equal to ò f .
n
n
n n E E
8. Suppose a sequence of measurable functions {f } defined on E converges pointwisely a.e.
n
on E to f. If|f | £ g on E for some integrable function g, then ò f n .............................. ò f .
n E E
31.3 Summary

 For a non-negative measurable function f : E  [0, ] (where E is a set which may be of
finite or infinite measure), we define
:
A ò f = sup{ A ò j j £ f on A, j Î B (E) }
0
for any A E.
Note that for non-negative bounded measurable functions vanishing outside a set of finite
measure, this definition agrees with the old one. Also note that we allow the functions to
take infinite value here.
 Suppose {f } is a sequence of non-negative measurable functions defined on E and {f }
n n
converges (pointwisely) to a non-negative function f a.e. on E. Then
E ò
E ò f £ lim inf f n
n
 If {f } is an increasing sequence of non-negative measurable functions defined on E
n
(increasing in the sense that f £ f for all n on E) and f  f a.e. on E, then
n n+1 n
E ò f ­ E ò f
n
by which it means {j f } is an increasing sequence with limit ò f .
E n E
 If {f } is a sequence of non-negative measurable functions on E, then
n
E ò liminf n f £ liminf n E ò f . The proof is easy and left as an exercise.
n
n
The following proposition is concerned with the absolute continuity of the integral.
 Suppose f is a non-negative measurable function defined on E such that f <  . Then for
E ò
all  > 0, there is a  > 0 such that
E ò f < 
whenever A  E with m(A) < .
 Suppose a sequence of measurable functions {f } defined on E converges pointwisely a.e.
n
E ò
on E to f. If|f | £ g on E for some integrable function g, then ò f n converges to f .
n E
31.4 Keywords

Fatou’s Lemma: Suppose {f } is a sequence of non-negative measurable functions defined on E
n
and {f } converges (pointwisely) to a non-negative function f a.e. on E. Then ò f £ lim inf f .
E ò
n E n
n
Monotone Convergence Theorem: If {f } is an increasing sequence of non-negative measurable
n
functions defined on E (increasing in the sense that f £ f for all n on E) and f  f a.e. on E, then
n n+1 n
E ò f ­ E ò f by which it means {j f } is an increasing sequence with limit ò E f .
n
E n
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