# Lebesgue measure and integration by P. K Jain, V. P. Gupta

By P. K Jain, V. P. Gupta

Released through John Wiley & Sons A Halsted Press booklet, 1986. This ebook offers a easy direction in Lebesgue degree, supplies designated reasons of purposes of labor and strategies used including explicitly in a simple, lucid sort. a really potent studying textual content.

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Sample text

2No( = c). This problem is yet unsolved and is called Cantor's continuum problem. However, many important results in mathematics have been proved by assuming that there is no cardinal number between No and c. This assumption is called the continuum hypothesis. Aocording to this hypothesis, c is the second transfinite cardinal number. Generalised Continuum Hypothesis. The assumption that, no matter what the infinite set A. is, there is no set with cardinal number p. h). K. h. is consistent with generally accepted axioms of the theory of sets, provided they are themselve\$ consistent.

Now, define a function/: N-+ AUB by /(n)= { an+I/2 if n is odd bn/2 if n is even. Clearly f defines a one-to-one correspondence. I 1. t. 8, case (ii). 2. The result can be extended to any finite number of sets. Remarks. 11 Corollary. The set I of all integers is countable. Proof. I Note. 2. 12 Theorem. q A and B are countable sets, then A x B is countable. Proof. If A or Bis empty, then the result is trivial. Assume that both A and B are nonempty sets. Since A and B are countable, we can write them as A= {a,: iEJ1} and B•{b1:jEJ2}, where J 1 and J2 are subsets of N.

Zo) :F [i/t(/2)](yo, zo), so i/i(fi) and i/i(/2) are different functions. Hence "1 is one-one. , ABxc. This gives the result in (b). (c) Here, we shall prove that (Ax B)c ...... AC x BC. Let fE(A x B)C. Then the values of the functions f are ordered pairs belonging to A x B. Thus we can write j(C)=(g(C), h(C)), where g(C)EA and h(C)EB. Therefore, gEAc and hEBc. e. an element of the set AC x Be. It can be easily verified that this correspondence is one-to-one. ,ACxBC. · /J)" = r1! ·{fl. 4 Theorem.