Page 374 - DMTH401_REAL ANALYSIS

Real Analysis

Notes 8. If A ’s are -algebras on a set X, then   := {A  X : A   for every } is also a -algebra
   
on X.
9. Show that () is generated by each of the following collections: {(a, ¥) : a  }, {[a, ¥) : a
 }, {(–¥, b) : b  }, {(–¥, b] : b  }, {(a, b) : a < b and a, b  }.

10. (i) If card(X)  card(), then card(X )  card(). (ii) If card(J)  card() and card(X ) 

card() for each   J, then, card(  X )  card(). [Hint: (i) Assume X = (0,1). Define a one-

J



one map f : (0, 1)  (0,1) as follows. If x = (x )  (0, 1) and if x = 0.x x , then f(x) =
n n n,1 n,2
2
0.x , x , x , x , x , x . (ii) Let g:   J and h :   X be surjections. Then f:    X
1 1 1 2 2 1 1 3 2 2 3,1    J 
2
defined as f(x, y) = h (x) is a surjection, and card( ) = card().]
g(y)
Answers: Self Assessment
1. Riemann integration theory 2. Lebesgue’s theory
3. d-dimensional Jordan outer content 4. Vitali set
5. continuous 6. monotone function
7. Borel -algebra 8. non-empty countable
30.8 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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