Let $(Omega,mathcal{F},mathbb{P})$ be a probability space, $a$ and $b$ be two rationals (i.e. $a$,$b$ $in mathbb{Q}$) such that $a<b$ and $(X_n)$ be a sequence of random variables defined on the above-defined measurable space. Set:

begin{equation}

Lambda_{a,b}={limsuplimits_{nrightarrowinfty}X_ngeq b; liminflimits_{nrightarrowinfty}X_n leq a}

end{equation}

begin{equation}

Lambda=bigcuplimits_{a<b} Lambda_{a,b}

end{equation}

Therefore, I “read” $Lambda$ as follows:

**there exists at least a pair of rationals $a$, $b$ ($a<b$) such that $limsuplimits_{nrightarrowinfty}X_ngeq b; liminflimits_{nrightarrowinfty}X_n leq a$**, that is

begin{equation}

Lambda={limsuplimits_{nrightarrowinfty}X_n> liminflimits_{nrightarrowinfty}X_n}

end{equation}

I am given that $mathbb{P}(Lambda_{a,b})=0$ $forall a, b in mathbb{Q}$ such that $a<b$. And, at this point, I know that one can state that

“$mathbb{P}(Lambda)=0$, since all rational pairs are countable”

I interpret the above statement in the following way, but I am not sure whether it is correct or not.

Since:

- $mathbb{P}(Lambda_{a,b})=0$ $forall a,b in mathbb{Q}$ such that $a<b$;
- $bigcuplimits_{a<b}Lambda_{a,b}$ is a countable union, since rationals are

countable by definition;

*by countable subadditivity property of probability measure $mathbb{P}$*, it follows that:

begin{equation}

mathbb{P}(Lambda)=mathbb{P}big(bigcuplimits_{a<b} Lambda_{a,b}big)leq sumlimits_{a<b}mathbb{P}big(Lambda_{a,b}big)=0

end{equation}

where $sumlimits_{a<b}mathbb{P}big(Lambda_{a,b}big)=0$ follows from the fact that I am given that $mathbb{P}(Lambda_{a,b})=0$ $forall a, b in mathbb{Q}$ such that $a<b$.

Hence,since probability lies by definition between $0$ and $1$, $mathbb{P}(Lambda)leq 0$ “means” that:

begin{equation}

mathbb{P}(Lambda)=0

end{equation}

*Is my reasoning correct?*