Hi context: i am trying to understand convergence of sequence of random variables. random variable are just measurable functions but I still cant get my head around the connection between sequence of functions and sequence of sets. To start suppose [tex]A_n \subset \Omega [/tex] i dont even understand this definition [tex] sup_{k \geq n} A_{k} := \bigcup^{\infty}_{k=n}A_k [/tex]. could someone explain this to me with a concrete example, or point me to a book that deals with sequence of sets and sequence of functions thanks
The definition you are concerned about simply says that if a point is any of the sets in the collection, it is in the sup, so the sup is then the union of all the sets in the collection. In other words, the sup is the smallest set containing all the points in any of the sets in the collection.
Uhm...I wrote a long post, but the program crashed...I'll try to recover some of it. First, you can check the first chapter of Sidney Resnick - A probability path, for the definitions of [tex] \limsup [/tex] and [tex] \liminf [/tex] Basically, the idea with those two animals is to capture the definition of a limit of a set...[tex] \limsup [/tex] and [tex] \liminf [/tex] always exist, but the limit of a set exists only when: [tex] \limsup A_n = \liminf A_n [/tex] (and, of course, the limit of the set is defined to be equal to both). If you check the definition for [tex] \limsup [/tex] the idea is to take the supremum of the sequence for each [tex] k \geq n [/tex] (which, as mathman said, is the smallest set from the group that contains all the elements) and the to try to bring it down, to check the limit from above. The idea with [tex] \liminf [/tex], on the other hand, is to take the limit from below. Check the definitions: For [tex] \limsup [/tex] you get the supremum with the unions and then you bring the set down with the intersections. It's analogous to what you do with integrals...you always have the upper and the lower integral -in which you approach the integral from above and from below-, but the integral exists only -and it is defined that way- when those two approaches coincide. Hope that helps, cd