Page 381 - DMTH401_REAL ANALYSIS

Unit 31: The Integral of a Non-negative Function

Lebsegue Dominated Convergence Theorem: Suppose a sequence of measurable functions {f } Notes
n
defined on E converges pointwisely a.e. on E to f. If|f | £ g on E for some integrable function g,
n
then f converges to f .
E ò
E ò
n
31.5 Review Questions
f
1. For a non-negative measurable function f defined on E, show that ò f = E ò c A for any A Í E.
A
Also show that ò f £ B ò f if A  B  E.
A
2. Show that if A, B C E are disjoint and f is a non-negative measurable function defined on
E, then ò f = A ò f + B ò f .
AÈ B
3. Show that if f is a non-negative measurable function defined on E and ò f = 0 , then f = 0
E
a.e. on E.
4. Show that if f is a non-negative measurable function defined on E and ò f <  , then f is
E
finite a.e.
5. Show that w may have strict inequality in Fatou’s Lemma.
(Hint: Consider the sequence {fn} defined by fn(x) = 1 if n x < n + 1,with fn(x) = 0 otherwise.)
6. Show that the monotone convergence theorem need not hold for decreasing sequence of
functions.
(Hint: Let fn(x) = 0, if x < n, fn(x) = 1 for xn.)
7. Show that if f and g are measurable and y |f| £ |g| a.e., and if g is integrable, then prove
that f is intergrable.

Answers: Self Assessment

:
1. A ò f = sup{ A ò j j £ f on A, j Î B (E) } 2. bounded measurable functions
0
3. non-negative function 4. increasing sequence
5. measurable function 6. A ò f £ A ò g

7. lim E ò g = E ò g <  8. converges to
n n

31.6 Further Readings

Books Walter Rudin: Principles of Mathematical Analysis (3rd edition), Ch. 2, Ch. 3.
(3.1-3.12), Ch. 6 (6.1 - 6.22), Ch.7(7.1 - 7.27), Ch. 8 (8.1- 8.5, 8.17 - 8.22).
G.F. Simmons: Introduction to Topology and Modern Analysis, Ch. 2(9-13),
Appendix 1, p. 337-338.

Shanti Narayan: A Course of Mathematical Analysis, 4.81-4.86, 9.1-9.9, Ch.10,Ch.14,
Ch.15(15.2, 15.3, 15.4)
T.M. Apostol: Mathematical Analysis, (2nd Edition) 7.30 and 7.31.

S.C. Malik: Mathematical Analysis.
H.L. Royden: Real Analysis, Ch. 3, 4.

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