What is the expansion of $\cap ^\infty _{m=1}\cup^\infty _{n=m}X_n$?

Question

It's the first time I'm seeing it and I'm having trouble constructing it inductively. I am guessing it's one of the two, but I am probably wrong:

$$X_1 \cap (X_1 \cup X_2)\cap (X_1\cup X_2\cup X_3)\cap...$$ $$X_1 \cap (X_1 \cup X_2)\cap ((X_1 \cap (X_1 \cup X_2))\cup X_4)...$$

Answer 1

Read from inside out.

$\bigcap_{m=1}^\infty \left(\bigcup_{n=m}^\infty X_n\right)$ is equal to:

$$(X_1\cup X_2\cup X_3\cup \dots) \cap (X_2\cup X_3\cup X_4\cup \dots) \cap (X_3\cup X_4\cup \dots) \cap \dots$$

It so happens that in the event that $X_n$ "converges" to a specific set $X$ then $\bigcap_{m=1}^\infty \left(\bigcup_{n=m}^\infty X_n\right) = X$ (for example in the event that we are talking about nested sets such as when $X_1\subseteq X_2\subseteq X_3\subseteq\dots$)

Answer 2

I don’t think it’s either of your suggestions, but it’s the set of all elements that appear in infinitely many $X_n$.

Answer 3

What you asked can also be found in the domain of probabilities:Consider $X_n$ to be a sequence of events and then $\bigcap_{m=1}^\infty \left(\bigcup_{n=m}^\infty X_n\right)$ will be the $limsupX_n$ If you study it there,you may be able to understand it better.