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Let [math](\Omega,\F,(\F_n)_{n\geq0},\p)[/math] be a filtered probability space. Let [math]X=(X_n)_{n\geq 0}[/math] be a supermartingale and let [math]T[/math] be a stopping time. Then

[[math]] X_T\in L^1(\Omega,\F,(\F_n)_{n\geq 0},\p) [[/math]]

and

[[math]] \E[X_T]\leq \E[X_0] [[/math]]

in each case of the following situations.

• [math]T[/math] is bounded.
• [math]X[/math] is bounded and [math]T[/math] is finite.
• [math]\E[T] \lt \infty[/math] and for some [math]k\geq 0[/math], we have
  [[math]] \vert X_n(\omega)-X_{n-1}(\omega)\vert\leq k, [[/math]]
  for all [math]\omega\in\Omega[/math].