The expected value of an everywhere stopped martingale (Q1082713)

Let \(\Omega\) be equal to C[0,\(\infty [\) or D[0,\(\infty [\), \(X_ t(\omega)=\omega (t)\), \(\omega\in \Omega\), \({\mathcal F}_ t=\sigma \{X_ s\), \(s\leq t\}\). Suppose that \((X_ t)\) is an (\({\mathcal F}_ t)\)-adapted martingale. Then for any stopping time \(\tau\) either \(EX_{\tau}=EX_ 0\) or \(EX_{\tau}\) is not defined, i.e. \(E[\max (X_{\epsilon},0)]=E[\max (-X_{\tau},0]=\infty\).