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Let $X_t$ and $Y_t$ two continuous stochastic process on $(Omega ,mathcal F,mathbb P)$ s.t. $X_t=Y_t$ a.s. for all $t$. Why $mathbb P{X_t=Y_ttext{ for all }t}=1$ ?

The solution goes as : We have that $X_r=Y_r$ a.s. for all $rinmathbb Q$. Therefore, $mathbb P{sup_{sinmathbb R}|X_s-Y_s|>0}=0$. The claim follow.

I really don't understand the argument. And for me $mathbb P(X_t=Y_t)=1$ for all $t$ is really the same as $mathbb P(X_t=Y_ttext{ for all }t)=1$. I don't see the difference.

The only thing you can say for $t$ fixed is $$mathbb P{X_t=Y_ttext{ for all }t}leq mathbb P{X_t=Y_t}.$$
You should pay more attention to the answers I gave you here. It's the same problem here.

What you must show is that there is $N$ with $mathbb P(N)=0$ s.t. for all $omega notin Omega $ and all $t$, $X_t(omega )=Y_t(omega ).$

• If $X_t=Y_t$ a.s. for all $t$, then $X_q=Y_q$ a.s. for all $qinmathbb Q$. Therefore, for all $qinmathbb Q$, there is $N_q$ with $mathbb P(N_q)=0$ s.t. for all $omega notin N_q$ $X_q(omega )=Y_q(omega )$.




• Set $N=bigcup_{qinmathbb Q}N_q$. It's s.t. $mathbb P(N)=0$ and for all $omega notin N$ and all $qinmathbb Q$, $X_q(omega )=Y_q(omega )$.




• Let $tin mathbb R$ and let $(q_n)_n$ a sequence of $mathbb Q$ s.t. $q_nto t$. If $omega notin N$, then, by continuity
  $$X_t=lim_{nto infty }X_{q_n}=lim_{nto infty }Y_{q_n}=Y_t.$$

Therefore $mathbb P{X_t=Y_ttext{ for all $t$}}=1$.

For a counter example for the equality : Let $mathbb P$ the Lebesgue measure on $Omega =[0,1]$. Let $tin [0,1]$, $X_t=0$ for all $omega $ and $Y_t(omega )=boldsymbol 1_{{t}}(omega )$.

Then $mathbb P{X_t=Y_t}=1$ for all $t$, but $$mathbb P{X_t=Y_ttext{ for all }t}=0,$$
since there is no $omega $ s.t. $X_t(omega )=Y_t(omega )$ for all $t$ because for $omega in Omega $ fixed, when $t=omega $ then $X_t(omega )=0$ but $Y_t(omega )=1$.

Since $X_t, Y_t$ are continuous, we have of course
$$
mathbb Pleft{sup_{sinmathbb R}|X_s-Y_s|=0right}
=
mathbb Pleft{sup_{sinmathbb Q}|X_s-Y_s|=0right}
=
mathbb Pleft(bigcap_{sinmathbb Q}{|X_s-Y_s|=0}right)
$$
a countable intersection of measurable sets of measure $1$.