Stopping Time for T

[Alexei]

If T, X_n are r.v. and A = {T >=n} is specified by X_n then is T a stopping time? How can this be proved?

 

[inferno]

I think it will not be a stopping time. rough idea:
{T=n}={T>=n}AND{T<=n}
={T>=n}ANDcomplement{T>=n+1}
{T>=n+1} is not Fn measurableble. it is F(n+1) measurable.

 

[inferno]

A refined logic:

Let T is stopping time. Then {T=n} is Xn measurable, {T=n-1} is X(n-1) measurable and hence Xn measurable and continuing in this way... {T=1} is Xn measurable.
Therefore {T=n}U{T=n-1}U...U{T=1}={T<=n} is Xn measurable. so, {T>=n+1} is Xn measurable which is a contradiction.