0.1 Set Theoretic Supremum & Infimum
Consider a sequence of sets \(\{ A_{n} \}_{n\in \mathbb{N}}\). In the context of the limsup and liminf of sets, the supremum of the sequence \(\{ A_{n} \}_{n\in \mathbb{N}}\) is the smallest possible upper bound for every element (set) of the sequence, i.e the smallest set containing each \(A_{n}\) which by definition is \[ \sup_{n \geq1}A_n= \bigcup_{n=1}^\infty A_n. \] Similarly, the infimum of the sequence \(\{ A_{n} \}_{n\in\mathbb{N}}\) is the largest possible lower bound of every element (set) of the sequence, i.e. the smallest collection of elements in each \(A_{n}\) which by definition is \[ \inf_{n \geq 1} A_{n}=\bigcap_{n=1}^\infty A_n. \]
0.2 Set Theoretic Limit Supremum & Infimum
Generally the limit supremum and limit infimum consider what happens to the supremum and the infimum as you disregard an increasingly larger finite collection of sets at the start of the sequence. So for the supremum we consider the union as we increase the lower bound \[ \limsup_{ n \to \infty } A_{n}=\lim_{ n \to \infty } \bigcup_{m=n}^\infty A_{m}. \] However, since the sequence \(\{ B_{n} \}_{n \in \mathbb{N}}\) where \(B_{n}=\bigcup_{m=n}^\infty A_{m}\) is monotone decreasing, this limit is equivalent to the intersection and so \[ \limsup_{ n \to \infty } A_{n}=\lim_{ n \to \infty } \bigcup_{m=n}^\infty A_{m}=\bigcap_{n=1}^\infty \bigcup_{m=n}^\infty A_{m}. \] Similarly, for the infimum we consider the intersection as we increase the lower bound \[ \liminf_{ n \to \infty } A_{n}=\lim_{ n \to \infty } \bigcap_{m=n}^\infty A_{m}. \] Again, since the sequence \(\{ B_{n} \}_{n \in \mathbb{N}}\) where \(B_{n}:=\bigcap_{m=n}^\infty A_{m}\) is monotone increasing, this limit is equivalent to the unions as so \[ \liminf_{ n \to \infty } A_{n} = \lim_{ n \to \infty } \bigcap_{m=n}^\infty A_{m}=\bigcup_{n=1}^\infty \bigcap _{m=n}^\infty A_{m}. \]
0.3 Limsup / Liminf & Probability Measure
An important notion in probability theory, specifically in the convergence of random variables and the Borel-Cantelli lemma is the notion of events occurring infinitely often which, somewhat counterintuitively, is equivalent to the notion of even limit supremum, i.e. for sequence of events \(A_{n}\), we have that \[ \mathbb{P}(\{ A_{n}\text{ occurs }i.o \}) = \mathbb{P}\left(\limsup_{ n \to \infty } A_{n}\right). \] A powerful tool for constructing proofs that events occur i.o. is the following theorem.
For a sequence of events \(\{ A_{n} \}_{n \in \mathbb{N}}\) we have that \[ \mathbb{P}\left(\liminf_{ n \to \infty } A_{n}\right) \leq \liminf_{ n \to \infty } \mathbb{P}(A_{n})\leq \limsup_{ n \to \infty }\mathbb{P}(A_{n})\leq \mathbb{P}\left( \limsup_{ n \to \infty } A_{n} \right). \]
\begin{proof} The second inequality is clear by definition hence we show the first (i) and third (iii) inequalities. For (iii) we have that \[
\limsup_{ n \to \infty } \mathbb{P}(A_{n})=\inf_{n\geq 1} \sup_{m \geq n}\mathbb{P}(A_{m})\stackrel{(\star)}{\leq} \inf_{n\geq 1}\sup_{m \geq n}\mathbb{P}\left( \bigcup_{k=m}^\infty A_{k} \right)=\inf_{n \geq 1}\mathbb{P}\left( \bigcup _{k=n}^\infty A_{k} \right)=\mathbb{P}\left( \bigcap_{n=1}^\infty \bigcup_{k=n}^\infty A_{k} \right)=\mathbb{P}\left( \limsup_{ n \to \infty }A_{n} \right) ,
\] where for \((\star)\) we used the monotonicity of measure and making use of the fact that \(\bigcup_{k=n}^\infty A_{k}\downarrow\) as \(n\uparrow\). For (i) we simply replace \(A_{n}\) with its complement \[
\limsup_{ n \to \infty } \mathbb{P}(A_{n}^c) \leq \mathbb{P}\left( \limsup_{ n \to \infty } A_{n}^c \right),
\] and then we obtain the result since \[
\mathbb{P}\left( \liminf_{ n \to \infty } A_{n} \right) =1-\mathbb{P}\left( \limsup_{ n \to \infty } A_{n}^c \right) \leq 1-\limsup_{ n \to \infty } \mathbb{P}(A_{n}^c)=\liminf_{ n \to \infty } (1-\mathbb{P}(A_{n}^c))=\liminf_{ n \to \infty } \mathbb{P}(A_{n}).
\] \end{proof}