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Measure Theory and Functional Analysis

Notes
(b) (f g) f g
E E E

(c) f 0 f 0 a.e.

(d) If f g a.e. then f g
E E

4. If f is integrable over E, then show that |f| is integrable over E, and f |f|.
E E

5. Show that if f is a non-negative measurable function then f = 0 a.e. on E iff f = 0.
E

6. If f = 0 and f (x) 0 on E, then f = 0 a.e.
E
11.5 Further Readings

Books Erwin Kreyszig, Introductory Functional Analysis with Applications, John Wiley &
Sons Inc., New York, 1989
Walter Rudin, Real and Complex Analysis, Third McGraw Hill Book Co., New York,
1987
R.G. Bartle, The Elements of Integration and Lebesgue Measure, Wiley Interscience,
1995

Online links www.maths.manchester.ac.uk
www.uir.ac.za

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