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Let X be a supermartingale. The function $t \mapsto E\{X_t\} $is right continuous iff there is exists a modification Y of X which is cadlag. Such a modification is unique.

I just beagn reading the book on Stochastic Integration by Protter and was looking for the proof of Theorem 9 which is stated above. Can somebody point me to a book or a place where I could find it? Thank you

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    That is Doob's regularization theorem. There is a proof of it in the book of Williams and Rogers (Diffusions, Markov processes, and martingales). – 2017-01-03
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    @Calculon Thanks a ton!! – 2017-01-03
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    You are welcome – 2017-01-03
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    @user3503589 : An alternative source is Karatzas and Shreve's book Brownian Motion and Stochastic Calculus or even the blog of George Lowther almost sure entry untitled "Cadlag Modifications". – 2017-01-04
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    @TheBridge I found the proof Thank you very much . – 2017-01-04

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