Page 387 - DMTH401_REAL ANALYSIS

Unit 32: The General Lebesgue Integral and Convergence in Measure

Self Assessment Notes

Fill in the blanks:
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1. A ............................. f is said to be integrable over E if f and f are both integrable over E.
2. Let g be integrable over E and let {f } be a sequence of measurable functions such that |f |
n n
g on E and for almost all x in E we have …………………….
3. Let {g } be a sequence of ……………….. which converges a.e to an integrable function g.
n
4. A sequence {f } of measurable functions is said to ……………….....….. in measure if, given
n
 > 0, there is an N such that for all nN we have m{x/|f(x) – fn(x)|} < .
5. Let {f } be a sequence of measurable functions that converges in measure to f. Then there is
n
a subsequence {nk f} that ................................ to f almost everywhere.
32.4 Summary

 Definition of General Lebesgue integral of a measurable function
 Properties of Lebesgue integral
 Lebesgue convergence theorem
 Generalization of Lebesgue convergence theorem
 Definition of convergence in measure of a sequence of measurable functions and
 Every sequence of measurable sequence that converges in measure contains a subsequence
that converges almost everywhere.
32.5 Keywords

Convergence in Measure: A sequence {f } of measurable functions is said to converge to f in
n
measure if, given  > 0, there is an N such that for all n N we have m{x/|f(x) – f (x)|} < .
n
Lebesgue Convergence Theorem: Let g be integrable over E and let {f } be a sequence of measurable
n
functions such that |f | g on E and for almost all x in E we have f(x) = lim f (x). Then
n n
ò
E ò f = lim f .
E n
32.6 Review Questions

1. Show that if f is integrable over E, then so is |f| and ò f £ E ò f . Does the integrability of
E
|f| imply that of f?.
2. Let {f } be a sequence of integrable functions such that f > f a.e with f integrable.
n n
f
Then f - ® if and only if f ® ò f .
0
ò
ò
n
n
3. Show that if f is integrable over E, then |f| is also integrable over E. further ò f £ E ò f is
E
the converse true?
Answers: Self Assessment
1. measurable function 2. f(x) = lim fn(x).
3. integrable functions 4. converge to f
5. converges

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