Let Γ be an open set and let X be adapted to {Ft}. Suppose that X is entirely continuous, or that X is a.s. continuous and  is complete. Show that HΓ need not be an   stopping time. Prove that Doob’s...

Let Γ be an open set and let X be adapted to {Ft}. Suppose that X is entirely continuous, or that X is a.s. continuous and
 is complete. Show that HΓ need not be an
  stopping time.

Prove that Doob’s L₂
inequality for one-dimensional cadllag martingales generalizes to continuous n-dimensional local martingales.