# If C 1, CZ, C3 . are sets such that Cr C Ch+ 1 . K = 1, 2, 3 . limk_ too CR is defined as the union C , UC 2 UC , U . Find lim k_ too CK. ( 2 ) CK =…

Please help with Probability Space functions. The problem is shown in the attachment. I was struggling in understand the concept and I’ll be grateful if you can show me the steps to solutions. Thanks in advance!

1 . If C 1, CZ, C3 . …. are sets such that Cr C Ch+ 1 . K = 1, 2, 3 . …. limk_ too CR is defined as theunion C , UC 2 UC , U …. Find lim k_ too CK.( 2 ) CK = Lac ER : 1 / K &lt; &lt; &lt; 3 – 1/ Kj , K = 1, 2, 3 , … .( b ) Ck = ( ( 2 , 4 ) ER 2 : 1 / k &lt; 2 2 + y 2 { 1 – 1 / K} , K = 1, 2, 3 . … .2 . If C 1 , C2, C3 . …. are sets such that ( * ) Ck+1 . K = 1 , 2 , 3 . … . limk_ too CR is defined as theintersection CIn Cz N C; M …. Find limk_ too CK.( a ) CK. = {` ER : 2 – 1 / K &lt; &lt; &lt; 2) , K = 1, 2, 3 . ….( b ) Ck = { ER : 2 &lt; &lt; &lt; 2 + 1 / K; ] , K = 1 , 2, 3 , … .( C ) Ck = ( ( 2 , 4 ) E RR 2 : 0 &lt; 2 2 + y ? &lt; 1 / K;] , K = 1, 2, 3 . ….3. For every one- dimensional set C , define the function @ ( C ) = &gt; of (a ) , where f (a ) – 12/ 31 ( 1/ 3 )2 .I = 0. 1, 2. …. zero elsewhere . If ( 1 = 12 : 2 – 0. 1, 2, 31 and C 2 – 12 : 2 – 0, 1, 2…. ], findQ ( ( 1 ) and Q ( ( 2 ) . Hint : Recall that S = at art … + arn- 1 = a ( 1 – pro ) / ( 1 – 8) , and hence , itfollows that Jim ~ _ too S = a / ( 1 – 7 ) provided that | ~| &lt; 1 .4. Let C be a set in one- dimensional space and let @ ( C ) be equal to the number of points in C’ whichcorrespond to positive integers . Then Q ( C ) is a function of the set C . Find @ ( C ) .( 2 ) C = PIER : 0 &lt; &lt; &lt; 5).( b ) C = 6 – 2, – 1} .( C ) C = LIER : – 00 &lt; &lt; &lt; 6}.