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I'm having problems to solve this problem

If X and Y are processes, Y is a modification of X and both have continuous trajectories a.s. Then X and Y are indistinguishable.

I know that if they are indistinguishable then are modifications of each other and on the other hand if they are modification aren't necessary indistinguishable, but I don't know how to use the Hypothesis of continuous trajectories a.s

Thank you in advance

stackQandA
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1 Answers1

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Since $X$ and $Y$ are modifications of eath other $P(X_t = Y_t)=1$ for all $t \in \mathbb{R}_+$. Since countable unions of nullsets are again nullsets, countable intersections of sets with full measure, have again full measure, so $P( \forall t \in \mathbb{Q}_+ : X_t=Y_t)=1$.

Since $X$ and $Y$ have continuous trajectories and $\mathbb{Q}$ is dense in $\mathbb{R}$ the conditions $\forall t \in \mathbb{Q}_+ : X_t=Y_t$ and $\forall t \in \mathbb{R}_+ : X_t=Y_t$ are equvialent.

user940347
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