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. – Jimmy R. Dec 30 '15 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.