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I learnt from measure theory that for a sequences of sets defined as $$\limsup A_n= \bigcap_{N=1}^\infty \left( \bigcup_{n\ge N} A_n \right)$$

and liminf is defined as $$\liminf A_n = \bigcup_{N=1}^\infty \left(\bigcap_{n \ge N} A_n\right)$$

then if they are equal we got $$\lim A_n = \limsup A_n =\liminf A_n $$

I want to ask that. Is there any attemp to defined this limit in usual way. As we have seen from sequences of real number or sequences of function. The statement "For each ..., there exist $N$ such that for all $k>N$,... "

Christopher
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1 Answers1

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This doesn't answer the question you asked, but perhaps this will help you think about it:

  • $\limsup$ is "infinitely often" (elements of $\limsup A_n$ are those contained in infinitely many of the $A_n$),
  • $\liminf$ is "eventually" (the elements of $\liminf A_n$ are those that are in every $A_n$ after a certain point).
yoyo
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    This is exactly the formal definition. In fact, if you translate your words into the language of for all and there exists (for the intersections and unions respectively), I believe you have an answer for OP. Since you have almost given an answer, I hope you will finish it! – Yuzuriha Inori Oct 19 '24 at 08:48