Let A 1 , A 2 , &#x2026; be a collection of events, and let A be the smallest &#x03C3; -field of subsets of &#x03A9; which contains all of them. If A &#x2208; A , then there exists a sequence of events { C n } such that C n &#x2208; A n &#xA0;and&#xA0; P ( A &#x25B3; C n ) &#x2192; 0 &#xA0;as&#xA0; n &#x2192; &#x221E; where A n is the smallest &#x03C3; -field which contains the finite collection A 1 , A 2 , &#x2026; , A n . However, I do not know how to choose the C n such that they turn up in &#x03C3; ( A 1 , . . A n ) ?

Question

Let           A              1              ,          A              2              ,    &amp;#x2026;   be a collection of events, and let           A       be the smallest     &amp;#x03C3;  -field of subsets of     &amp;#x03A9;   which contains all of them. If     A    &amp;#x2208;          A      , then there exists a sequence of events           {              C                  n                    }       such that              C              n              &amp;#x2208;                  A                    n                  &amp;#xA0;and&amp;#xA0;          P              (      A      &amp;#x25B3;              C                  n                    )        &amp;#x2192;    0        &amp;#xA0;as&amp;#xA0;    n    &amp;#x2192;    &amp;#x221E;      where                   A                    n             is the smallest     &amp;#x03C3;  -field which contains the finite collection           A              1              ,          A              2              ,    &amp;#x2026;    ,          A              n              .      However, I do not know how to choose the           C      n       such that they turn up in     &amp;#x03C3;    (          A      1        ,    .    .          A      n        )  ?

Elias Flores · Accepted Answer

Union of             A        n  &#039;s is an algebra which generates       A  . Just apply Halmos&#039; Theorem now. You get       C    n    ∈            A                      k        n             for some       k    n   incerasing to   ∞. But you can keep repeating each       C    n   to make sure that       C    n    ∈            A        n   for each   n: Look ar   ∅  ,  ∅  ,  ∅  ,  .  .  .  ,      C    1    ,      C    1    ,  .  .  .      C    1    ,      C    2    ,      C    2    ,  .  .  .  ,      C    2    ,  .  .  . where   ∅ is repeated       k    1    −  1 times,       C    1   is repeated       k    2    −      k    1    −  1 times, and so on.

Let A <mrow class="MJX-TeXAtom-ORD"> 1 </mrow> </msub> , A <mro

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