1

I was scrolling through a textbook on probability theory (using measure theory) and one of the excercises states:

Let $\mathcal{A}$ be a $\sigma$-algebra and $(A_n)_{n\geq1}$ a sequence of events in $\mathcal{A}$. Show that $$\underset{n\to\infty}{\lim\inf} \;A_n\in\mathcal{A}, \quad \underset{n\to\infty}{\lim\sup} \;A_n\in\mathcal{A},\quad\underset{n\to\infty}{\lim\inf} \;A_n \subset \underset{n\to\infty}{\lim\sup} \;A_n$$

What does it mean for a sequence of events (the subsets $A$ of the $\mathcal{A}$ $\sigma$-algebra) to take its limit (as $n$ approaches infinity), inf and sup?

I am not looking for a proof of the statements in question (even though a link to a source would be interesting as well).

Asaf Karagila
  • 393,674
math mat
  • 91
  • https://math.stackexchange.com/questions/172167/interpretation-of-limsup-liminf-of-sets – stoic-santiago Nov 17 '21 at 16:59
  • See also https://math.stackexchange.com/q/107931/215011 – grand_chat Nov 17 '21 at 16:59
  • It turns out $\liminf A_n = { x \mid x \in A_n \text{ for all but finitely many $n$} }$ and $\limsup A_n = { x \mid x \in A_n \text{ for infinitely many $n$} }$. You can read more about why this is true at an old blog post of mine here – HallaSurvivor Nov 17 '21 at 21:15

0 Answers0