TITLE: A continuum of independent random variables
QUESTION [8 upvotes]: How can one construct a probability space that supports a continuum of independent random variables?

REPLY [3 votes]: To elaborate on Chris's hint, it follows from KET that as soon as you define finite-dimensional distributions which satisfy two simple properties, you can construct a measure on the product space which have those finite-dimensional distributions as marginals. Now, let us consider $\xi_t$ be a random variable for $t\in[0,1]$ where $\xi_t$ has a distribution $\mu_t$  and $\xi_t\perp\xi_s$ whenever $s\neq t$. Then
$$
  \nu_{t_1,\dots,t_n}(A_1\times \dots\times A_n) := \prod_{i=1}^n\mu_{t_i}(A_i)
$$
are the correspondent finite-dimensional distributions. You can easily check that they satisfy all the required properties needed to apply KET.