Expectation of X_t When Pr(X_t>b)=0: Finite?

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If i know that Pr(X_t>b)=0, where X_t>0 and b is positive finite, then should the expectation of X_t be finite? Is there any case where it is infinite?
 
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Yes. Proof is trival. 0<=X _t <=b implies E(X_t)<=b.
 

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