Page 164 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 12: General Convergence Theorems

Uniform Convergence, Almost Everywhere (a.e.): Let <f > be a sequence of measurable functions Notes
n
defined over a measurable set E. Then the sequence <f > is said to converge uniformly a.e. to f,
n
if a set E E s.t.
0
(i) m (E ) = 0 and
0
(ii) <f > converges uniformly to f on the set E – E .
n 0
12.4 Review Questions

1. Show that we may have strict inequality in Fatou’s Lemma.
2. Let <f > be an increasing sequence of non-negative measurable functions, and let f = lim f .
n n
Show that f lim f n .

Deduce that f u , if u is a sequence of non-negative measurable functions and
n n
n 1
f u .
n
n 1
3. State the Monotone Convergence theorem. Show that it need not hold for decreasing
sequences of functions.
4. Let {g } be a sequence of integrable functions which converge a.e. to an integrable function
n
g. Let {f } be a sequence of measurable functions such that |f | g and {f } converges to f
n n n n
a.e.
If g lim g
n n

then prove that f lim f .
n
n
5. State and prove monotone convergence theorem.

12.5 Further Readings

Books G.F. Simmons, Introduction to Topology and Modern Analysis, New York: McGraw
Hill, 1963.
H.L. Royden, Real analysis, Prentice Hall, 1988.

Online links dl.acm.org
math.stanford.edu
www.springerlink.com

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