What almost sure convergence means in the context of strong law of large numbers According to http://en.wikipedia.org/wiki/Almost_sure_convergence#Almost_sure_convergence, a sequence  of random variables $X_n$, which are a function of a shared sample space $Ω$, is said to converge almost surely to $X$ when:
$
    \operatorname{Pr}\Big( \omega \in \Omega : \lim_{n \to \infty} X_n(\omega) = X(\omega) \Big) = 1.
  $
The strong law of large numbers says the sample average converges almost surely to the expected value:
$
    \overline{X}_n\ \xrightarrow{a.s.}\ \mu \qquad\textrm{when}\ n \to \infty.
  $
I  am confused as to what this means. For every $n$, $\overline{X}_n$ has a different sample space - the cartesian product corresponding to $n$ i.i.d instances of $X$. How can the definition of almost sure convergence apply here? What is the shared sample space?

 A: Certainly, in order for the SLLN to be non-vacuous, we need to have an appropriate shared sample space where all the $X_n$ are defined.  (Given this, all of the $\overline{X}_n$'s are defined on that same space.)

Theorem. Given any probability distribution function (or Borel probability measure) $F$ on $\mathbb{R}$, there exists a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and a sequence of measurable functions $X_1, X_2, \dots : \Omega \to \mathbb{R}$, such that the $X_n$ are iid random variables with common distribution $F$.  Moreover:

*

*$\Omega$ can be taken to be the countable Cartesian product $\mathbb{R}^{\mathbb{N}}$, or equivalently the set of all sequences of real numbers;


*$\mathcal{F}$ can be taken to be the product $\sigma$-algebra on $\mathbb{R}^{\mathbb{N}}$, which is generated by all "cylinder sets" of the form $A_1 \times A_2 \times \dots \times A_n \times \mathbb{R} \times \mathbb{R} \times \cdots$, where $n$ is any integer and $A_1, \dots, A_n \subset \mathbb{R}$ are Borel;


*$X_n$ can be taken to be the coordinate function $X_n(x_1, x_2, \dots) = x_n$.

The measure $\mathbb{P}$ on $\mathbb{R}^{\mathbb{N}}$ is sometimes called "infinite product measure".  It can be constructed directly using the Caratheodory extension theorem, but is usually treated instead as a special case of the more general Kolmogorov extension theorem, which allows for dependence between the $X_i$.  The Kolmogorov extension theorem is treated in most measure theory texts, such as for instance these notes by Terry Tao.
