Exercise
45
[0BP] Let \(A_ 1,A_ 2\ldots \) be sets , for \(nββ\); let \(X=β_ n A_ n\). We define the characteristic function \({\mathbb 1}_ A: Xββ\) as
\[ {\mathbb 1}_ A(x)= \begin{cases} 1 & \text{if}~ xβ A\\ {} 0 & \text{if}~ xβ A \end{cases} ~ . \]
We will use the definitions \(\limsup _{n} A_ n\) and \(\liminf _{n} A_ n\) seen in eqn.Β [(3.286)] and [(3.287)]. You have
\begin{eqnarray} \label{eq:limsup_ insiemi_ 1} {\mathbb 1}_{(\limsup _{n} A_ n)} & =& \limsup _{n} {\mathbb 1}_{A_ n} ~ ~ ,\\ {} \label{eq:liminf_ insiemi_ 1} {\mathbb 1}_{(\liminf _{n} A_ n)} & =& \liminf _{n} {\mathbb 1}_{A_ n} ~ ~ . \end{eqnarray}