Consider the probability space where $=[0,1]\backslash$times $[0,1],$ F $=\backslash$sigma $(\{[$a$,$b$]\backslash$times $[$c$,$d$]$ : a$,$b$,$ c$,$d in $[0,1]\}),$

and the product measure P with the property that P$[[$a$,$b$]\backslash$times $[$c$,$d$]]=($b$$a$)($d $$c$)$ whenever

a $<=$ b$,$ c $<=$ d$.$

Consider the following subset of $$: E $=\{(\backslash$omega $1,\backslash$omega $2)$ in $$ : $\backslash$omega $1$ in $[0,1],\backslash$omega $2=\backslash$omega $1\}.$

$($a$)$ Why is E an event $($i$.$e$.,$ E in F$)?$

$($b$)$ What is the value of P$[$E$]?$