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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.

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  • $\begingroup$ 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. $\endgroup$ – Jimmy R. Dec 30 '15 at 14:49
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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.

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