Is $P(X_t=Y_t,\forall\ 0\leq t < \infty )$ well defined? I have an elementary question about stochastic processes (continuous) and the notion of two processes being indistinguishable. Let $X=(X_t)_{t\geq 0}, Y=(Y_t)_{t\geq 0}$ be two stochastic processes on the same probability space $(\Omega, \mathcal{F}, P)$. I have the following definition, from Karatzas, Shreve - Brownian Motion and Stochastic Calculus.

Now there is this another definition, which comes from Kannan, Lakshmikantham - Handbook of Stochastic Analysis with Aplications.
 
Basically it says that there is a measurable set $A\in\mathcal{F}$ such that $P(A)=1$ and $X_t(\omega)=Y_t(\omega)$ for all $\omega\in A$ and all $0\leq t < \infty$. I've seen both definitions in different kind of formats, for example, this second definition is equivalent to this one, from Achim Klenke- Probability Theory, A Comprehensive Course.
 
In this case, he is writing $\mathcal{A}$ instead $\mathcal{F}$. 
The last two definition are well posed but the first one it's not, because   $$\{X_t=Y_t,\forall\ 0\leq t < \infty\} = \bigcap_{0\leq t < \infty}\{X_t=Y_t\}$$ 
is an uncountable intersections of measurable sets, and we know that this is not necessarily a measurable set. Therefore, if someone writes $P(X_t=Y_t, \forall\  0\leq t < \infty )$, then he/she is just hoping to have a $\sigma$-algebra rich enough to comport this set or he/she is just wrong (because they want to measure a non-measurable set).
On the other hand, the last two definition (in special, the definition from Kannan's) makes we think in a way that lead us to the first (and not necessarily well posed) definition. In fact, with that set $A\in\mathcal{F}$ defined above, we can kind of say $A = \{\omega\in\Omega: \ X_t(\omega)=Y_t(\omega),\forall\ 0\leq t <\infty \}$, maybe not correct in some points of a null measure set. Now we can kind of write $P(A) = P(X_t=Y_t,\forall\ 0\leq t < \infty)$. Then we get the first definition, which would be correct, except, maybe, in a null measure set.
I have 2 interpretations of this situation, and I hope you can help me to have a correctly interpretation (maybe all my interpretations are wrong):
1) The first definition is in fact just a notation that we use when the conditions from the second definition are satisfied;
2) The first definition is not a notation, it means exactly this, but it's implied on the definition that the set $\{X_t=Y_t,\forall\ 0\leq t < \infty\}$ is measurable.
Thank you for the help. 
 A: In view of whuber's comments, I will post an answer. I think this settled the question and I don't feel this definition is strange anymore.
Saying that $X,Y$ will be indistinguishable if $P(X_t=Y_t,\forall\ 0\leq t < \infty) = 1$ was strange to me because we are asking for uncountable intersection to be measurable. So we may try weaken the definition saying that $X,Y$ will be indistinguishable if exists a measurable set $A\in\mathcal{F}$ such that $P(A) = 1$ and $X_t = Y_t$ in $A$ for all $0\leq t < \infty$.
It's true that $A\subset\{\omega\in\Omega: \ X_t(\omega)=Y_t(\omega), \forall\ 0\leq t < \infty \}$, but it may not be equal. Therefore there is a null set $N\subset\Omega$ such that $A\cup N = \{\omega\in\Omega: \ X_t(\omega)=Y_t(\omega), \forall\ 0\leq t < \infty \}$.
As pointed out by whuber in the comentns, the probability space is supposed to be complete. So we have that $N$ is measurable, with $P(N) = 0$. In this case, we can just include $N$ in the definition and use $A\cup N$ instead of $A$. Therefore, the weak definition implies the strong, thus the definitions are equivalent.
PS: by a null set I mean a set contained in a measurable set with zero measure. Maybe the latter is called null set, I'm not sure. This is irrelevant when the space is complete, but I wanted to highlight the fact that this is relevant when the space is not complete. 
A: The "for all" suggests an intersection as opposed to a union.
$$
\{X_t = Y_t; \forall 0 \leq t < \infty\} = \bigcap_{t = 0}^{\infty} \{X_t = Y_t \}.
$$
