According to page 3 of J. Jacod and A. Shiryaev's Limit Theorems for Stochstic Processes (2013),
Note that if $X$ and $Y$ are indistinguishable, one has $X_t = Y_t$ a.s. for all $t \in \mathbb{R}_+$, but the converse is not true. This converse is true, however, when both $X$ and $Y$ are càd, or càg.
Also, they define indistinguishablility in 1.10 of their book:
1.10 A random set $A$ is called evanescent if the set $\{ \omega: \exists{t \in \mathbb{R}_+} \text{ with }(\omega, t) \in A \}$ is $\mathbb{P}$-null; two $E$-valued processes $X$ and $Y$ are called indistinguishable if the random set $\{ X \neq Y\} = \{ (\omega, t): X_t(\omega) \neq Y_t (\omega) \}$ is evanescent, i.e. if almost all paths of $X$ and $Y$ are the same.
However, I'm confused that $X_t = Y_t $ a.s. for all $t \in \mathbb{R}_+$ does not imply indistinguishability.
Is there any example to show that the converse is not true?
Here, what I have done.
"$X_t = Y_t $ a.s. for all $t \in \mathbb{R}_+$" exactly means
\begin{align*}
&\mathbb{P} \left\{ X_t = Y_t \text{ for all } t \in \mathbb{R}_+ \right\} = 1 \\
\iff & \mathbb{P} \left\{ X_t = Y_t \text{ for all } t \in \mathbb{R}_+ \right\}^c = 0 \\
\iff & \mathbb{P} \left\{ \exists{t \in \mathbb{R}_+}\text{ such that } X_t \neq Y_t \right\} = 0.
\end{align*}
Since the event set $\left\{ \exists{t \in \mathbb{R}_+}\text{ such that } X_t \neq Y_t \right\}$ is the same for $\{ \omega: \exists{t \in \mathbb{R}_+} \text{ with } X_t(\omega) \neq Y_t(\omega) \}$, I think they should be equivalent.
Is there any mistake that I have?
For example, I may misunderstood the statement "$X_t = Y_t $ a.s. for all $t \in \mathbb{R}_+$" which is actually meaning that for every fixed $t \in \mathbb{R}_+$, $X_t = Y_t$ a.s. for each.
In this case, the indistinguishability is "$X_t = Y_t $ for all $t \in \mathbb{R}_+$ a.s.".
Thanks,
Best Answer
My definition of indistinguishable for $X,Y$ is $P(\exists t \in T : X_t \ne Y_t) = 0$. More precisely is $\exists N \in \mathcal{A} : P(N) = 0$ and $\bigcup\limits_{t \in T} \{X_t \ne Y_t\} \subseteq N$.
An other definition I have is : $Y$ is a modification of $X$ if $\forall t \in T, \hspace{0.1cm} X_t = Y_t \hspace{0.1cm} P-a.s$
What is true is that $X,Y$ being indistinguishable implies that $X$ is a modification of $Y$ since $\forall t \in T \{ X_t \ne Y_t \} \subseteq \{\exists t \in T : X_t \ne Y_t\}$.
That being said the converse is not true, i.e modification doesn't imply indistinguishable, since you have the conterexample with $\Omega = ([0,1], \mathcal{B}(\mathbb{R}),\mathcal{L})$ and $X(x,t) = \begin{cases} 1 & x = t \\ 0 \end{cases}$ and $Y(x,t) \equiv 0$
I don't know wheter this answers you question, hope it helps.