Consider the measurable space (W, A) and let A n A, n 1, 2, then recall that And (i) Show that A n A n (If also A n A n so that A n n A n then this set is denoted by A n and is called the limit of the sequence A n , n 1 (ii) Show that ( A n ) n A c n ( A n )c n A c n conclude that if lim n A n A, t...

i x A n if and only if x Z n 1 j n A j so that x j n 0 A j for some n 0 1 and then x A j for all j n 0 or x j n A j for all n 1 so that x n 1 j 1A j nA n ii n 1 j n Ac j n A c n n A n c n 1 j n A c j ...

Consider the measurable space (W, A) and let A n A, n = 1, 2, then

Question:

Consider the measurable space (W, A) and let A_nˆˆ A, n = 1, 2,€¦ then recall that

lim inf A, = lim A, =UNA n=1 j=n %3D n 00

And

Consider the measurable space ((, A) and let An ˆˆ

(i) Show that A_n A_n. (If also A_n A_n€™so that A_n = _n†’ˆžA_n€™ then this set is denoted by A_nandis called the limit of the sequence {A_n}, n ‰¥ 1.

(ii) Show that ( A_n) = _n†’ˆžA^c_n( A_n)c = _n†’ˆžA^c_nconclude that if lim _n†’_¥ A_n = A, then lim _n†’_¥ A^c_n = A^c.

(iii) Show that

Consider the measurable space ((, A) and let An ˆˆ

And

(iv) Show that

And

(v) By a counterexample, show that the inverse inclusions in part (iv) do not hold, so that _n†’ˆž (A_n Ãˆ B_n) need not be equal to (_n†’ˆžA_n) Ã‡ (_n†’ˆž B_n), andA_n(A_n Ãˆ B_n) need not be equal to (A_n) Ãˆ (B_n).

(vi) If lim_n†’_¥ A_n= A and lim_n†’_¥ B_{n =}B_,then show that lim_n†’_¥ (A_n Ã‡ B_n) = A Ã‡ B and lim_n†’_¥ (A_n Ãˆ B_n) = A Ãˆ B.

(vii) If lim_n†’_¥ A_n = A, then show that for any set B, lim_n†’_¥ (A_n DB) = ADB. where A_n D B is the symmetric difference of A_n and B.

(viii) If A_2j_€“1 = B and A_2j = C, j = 1, 2, €¦, determine A_nand n†’ˆžA_n. Under what condition on B and C does the limit exist, and what is it equal to?