Stochastic processes equal in norm are modifications of eachother.

Question

We can define a normed space of processes by introducing the norm

$\mathbb{E}\int_{0}^{T}\mid X(t)\mid dt<\infty$.

This means that

$\int_{0}^{T}\int_{\Omega}\mid X(t)\mid dP dt$, hence if $||X-Y||=0$

we get that

$\int_{\Omega}\mid Y(s)- X(s)\mid dP=0$

Thus $Y=X$ a.s for all $t$, which means that the processes are modifications(https://en.m.wikipedia.org/wiki/Stochastic_process#Modification) of eachother.

Are my implications correct?

Answer 1

What if $X(t)=0$ for all $t$ and $Y(t)=0$ for $t≠1, Y(t)=1$ for $t=1$?