logged in for the first time since forever lol. SO MUCH HAS CHANGED! i have a paper cooking! on the connectivity of random graphs with a fixed degree sequence. some very new and (in my very biased opinion) great results
"composition of measurable functions is not necessarily measurable" is misleading bc it's only true for lebesgue measurable i.e. preimage of borel sets being lebesgue sets, where there's a mismatch between the sigma algebra used on the domain vs the image.
tau is an optional time wrt Ft+ iff it is an optional time wrt Ft-
proof:
"<=" trivial (Ft- is in Ft+)
"=>": note union {tau<t-1/n} = {tau<t}, and each {tau<t-1/n} is in F(t-1/n}+ in Ft-
corollary: tau being an optional time wrt Ft+ or Ft- or Ft are equivalent
broke: a stopping time is an optional time under the same filtration
enlightened: knowing whether it's too early is stronger than knowing whether it's too late (since too early => not too late, and not too early but too early up to now => not too late, & too late otherwise)
