$\lim \sup\{X_n\geq x\}$ vs $\{\lim \sup X_n \geq x\}$

Question

Let $(X_n)$ (n is a natural number) be a sequence of real valued random variables. For any real number $x$, let's define: $E_x = \limsup \{ X_n \geq x\} $, $F_x = \{\limsup X_n \geq x\} $

If $x$ is fixed, is one of the following Relations valid (why?): $E_x \subset F_x$, $E_x \supset F_x$ or $E_x=F_x$

What if, for example, $a < b$ (for $E_a$, $F_b$)?

Answer 1

For the sake of having an answer:

• $E_x\subset F_y$ is guaranteed if and only if $y\leqslant x$.
• $F_x\subset E_y$ is guaranteed if and only if $y\lt x$.