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It is easy to give examples of non-separable $L^p$ spaces by considering a measure on a big space. If one adds the condition that the space has to have total measure 1, the problem is not as easy.

The following equivalences are known, and not very useful:

  1. $L^1(\Omega,A, P)$ is separable

  2. for all $p \in [1,\infty)$, $L^p(\Omega,A,P)$ is separable

  3. the $\sigma$-algebra $A$ of measurable sets with the pseudometric $\rho(a,b)=P(a\Delta b)$ is separable, where $\Delta$ is the symmetric difference.

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    $\begingroup$ Perhaps an uncountably infinite product of probability measures will do the trick; search for infinite product measure in Google. $\endgroup$
    – Philip Brooker
    Commented Jun 16, 2011 at 21:28

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Philip is right. A family of events is independent if any finite subfamily is independent. In order to discuss an uncountable independent family of events, we need a probability space that is not separable in your pseudometric, or equivalently the $L^1$ space is not separable. As usual, we may model the situation by taking a product of copies of $\{0,1\}$, with one factor for each of the events. And then use the (uncountable) product measure.

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