Sheet1 - ( B ) ) = B Provide counterexamples for each of...

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Statistics 330/600 Due: Thursday 20 January 2005: Sheet 1 Please attempt at least the starred problems. *(1.1) Please print clearly your name, year, and major or graduate department. *(1.2) let R denote the set of all half-open rectangles of the form ( a 1 , b 1 ] × ( a 2 , b 2 ]in R 2 . Show that R generates the Borel sigma-Feld B ( R 2 ) . (1.3) Suppose T maps a set X into a set Y o r B Y deFne T 1 B : ={ x X : T ( x ) B } o r A X deFne T ( A ) : ={ T ( x ) : x A } . Some of the following assertions are true and some are false. T ( i A i ) =∪ i T ( A i ) and T 1 ( i B i ) =∪ i T 1 ( B i ) T ( i A i ) =∩ i T ( A i ) and T 1 ( i B i ) =∩ i T 1 ( B i ) T ( A c ) = ( T ( A )) c and T 1 ( B c ) = ( T 1 ( B )) c T 1 ( T ( A ) ) = A and T
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Unformatted text preview: ( B ) ) = B Provide counterexamples for each of the false assertions. (1.4) The set R = {} R {} is called the extended real line . Write A for the sigma-Feld on R generated by B ( R ) together with the two singleton sets {} and {} . (i) Show that A is also generated by E = { [ , t ] : t R } . (ii) Show that A equals the Borel sigma-Feld for the metric d ( x , y ) : = | arctan ( x ) arctan ( y ) | on R ....
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