A few days ago, I was introduced the concept of convergence in probability, after the almost sure convergence.

I understood the almost sure convergence (I think):

We have a sequence of random variables $$\{X_n\}_{n\in\mathbb{N}}$$ that "goes" to a fixed one $$X$$, but, maybe, there are sets with measure zero where it's not true. We don't care, and we say that the $$\{X_n\}_{n\in\mathbb{N}}$$ converges almost surely to $$X$$.

Now, they introduce to me the convergence in probability:

$$\{X_n\}_{n\in\mathbb{N}}$$ converges in probability to X if $$P\{\omega\in\Omega\mid\ \! \! \!\mid X_n(\omega)-X(\omega)\mid >\epsilon \}\to0$$

as $$n$$ goes to infinity.

After that, they say that if $$\{X_n\}_{n\in\mathbb{N}}\to X$$ in probability and $$\{X_n\}_{n\in\mathbb{N}}\to Y$$ in probability, then $$X=Y$$ almost surely.

My question is this:

Let's assume that there are $$X$$ and $$Y$$ random variables that are equal almost surely (that is, in every set of measure different of $$0$$ they are equal). It means that there are some $$\omega \in \Omega$$ where $$X(\omega)$$ exists, but $$Y(\omega)$$ don't (or inverse)? Or $$Y$$, $$X$$ and every $$X_n$$ has to be defined (it means, take values) for the same $$\omega \in \Omega$$? If the answer to the second question is yes, then I think that I understand the definition, but if it's no, then the definition takes $$\omega$$ that could have sense in $$X_n$$, but not in $$X$$.

Thanks for taking the time.

• Restrict attention to $$Ω \setminus \left\{N(X)\cup N(Y)\cup N(X_n)\right\}$$ where $N(X), N(Y), N(X_n)$ are all sets of measure zero (hence their union has also measure zero) and are exactly the sets at which all of these problems may occur. Commented Dec 30, 2015 at 14:49

The usual definition of a random variable (for example Wikipedia) is as a measurable function whose domain is $\Omega$. In particular, this is a function, so it is defined for all $\omega\in\Omega$. So yes, $X,Y,X_n$ should all be defined for all $\omega\in\Omega$.
Since for probabilistic purposes sets of measure zero don't matter, you could define a random variable as a measurable partial function on $\Omega$, which is defined except for a set of measure zero. In this case $X,Y,X_n$ may not always be defined, but, as in Stef's comment, they will still all be defined almost everywhere, so convergence in probability still makes sense.