hw5add - H as follows: if A = Q \ I 2 H , then m ( A ) = b...

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1. (a) Prove that, if a subset E in metric space ( ) is of the &rst cat- egory and A E , then A (b) Prove that if ( E n ) 1 n =1 ( ) category, then E = [ 1 n =1 E n 2. (a) Prove that, if a subset E in metric space ( ) is of the &rst cat- egory and A E , then A (b) Prove that if ( E n ) 1 n =1 ( ) category, then E = [ 1 n =1 E n 3. Consider the metric space ( N ) , where N is the set of natural numbers and : N ± N ! R ( m;n ) = & 0 if m = n; 1 + 1 m + n if m 6 = n: Prove: (a) ( N ) is a metric space. (b) The metric space ( N ) is complete . (c) Let B n = B N ± n; 1 + 1 2 n ² = m 2 N j ( n;m ) ² 1 + 1 2 n ³ . Prove that B 1 ³ B 2 ³ ::: ³ B n ³ ::: and \ 1 n =1 B n = ? . 4. Let X = Q H = f X \ I j where I = ( a;b ) ; [ a;b ] ; ( a;b ] ; [ a;b ) ; ´1 < a ² b < + 1g : (a) Prove that H is a semi-ring.
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Unformatted text preview: H as follows: if A = Q \ I 2 H , then m ( A ) = b a if I = ( a;b ) ; [ a;b ] ; ( a;b ] or [ a;b ) : Prove that m is a measure. (c) Prove that the measure m on the semi-ring H is not sigma-additive. 1 Remark : It is assumed that you know the notion of connected set: a set A in a metric space ( M;&amp; ) is called disconnected if there are open sets U;V &amp; M such that (i) A &amp; U [ V; (ii) U \ V = ? ; (iii) A \ U 6 = ? ;A \ V 6 = ? . A connected set is the set which is not disconnected. You can use the theorem stating that a set in R is connected if and only if it is an interval (bounded, unbounded, open, semi-open or closed). 2...
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hw5add - H as follows: if A = Q \ I 2 H , then m ( A ) = b...

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