# Interchangeability of $\limsup$

Let $(X_n)$ be a sequence of real random variables on some probability space $(\Omega,\mathcal{F},P)$ and let $a\in\mathbb{R}$.

Note that

\begin{align} \limsup_n\,[X_n\geq a] &= \bigcap_{n}\bigcup_{k\geq n}[X_k\geq a]\\ &= \{\omega\in\Omega\colon \forall_n\exists_{k\geq n}X_k(\omega)\geq a\}\\ &= \{\omega\in\Omega\colon \forall_n\sup_{k\geq n}X_k(\omega)\geq a\}\\ &= \{\omega\in\Omega\colon \inf_n\sup_{k\geq n}X_k(\omega)\geq a\}\\ &= [\limsup_n X_n\geq a]\\ \end{align}

(Is this even correct?)

When can we change the $\limsup$ "into the set"? It seems to work for functions on the left side of $\geq$ as then $\inf_n$ translates to "and" and $\sup$ translates to "or".

I really like the analogies $\bigcap\leftrightarrow\forall$ and $\bigcup\leftrightarrow\exists$. Is there a more general framework for extending these to $$\bigcap\leftrightarrow\inf\leftrightarrow\forall\quad\text{and}\quad \bigcup\leftrightarrow\sup\leftrightarrow\exists?$$

• Typo - first equation, right side - should be $X_k\ge a$. Major question - the first line has a lim sup of a random variable compared to a number. Do you mean it for all realizations of$X_n(\omega)$? – herb steinberg Jul 8 '18 at 15:09
• @herbsteinberg Thank you, edited. Answering your question: To me, $[X_n\geq a]$ is the set of all $\omega\in\Omega$ s.t. $X_n(\omega)\geq a$. So, in the first line, I am speaking of a $\limsup$ of sets. – Ramen Jul 8 '18 at 15:17
• It looks to me to be obvious in a way. You want the equation to hold for all$\omega$. However as you have, it will hold for each $\omega$ individually, so it follows that it holds for all. Am I missing something? – herb steinberg Jul 9 '18 at 16:49
• @herbsteinberg Yes, I think the equation is true, too. I rather stated it as an example leading to the second part of the question, that is I am mainly interested in how one could generalise the situation. I could imagine it leading to lattices or something similar? – Ramen Jul 9 '18 at 18:07
• I don't really understand the second part. "into the set"? – herb steinberg Jul 9 '18 at 23:41