This is should probably rather be a comment to Michael Greinecker's answer, but I do not have the necessary privileges.

Michael Greinecker's answer leaves open what happens with a continuum-sized discrete space when one does *not* assume the continuum hypothesis.

Arnold W. Miller showed in section 4 of [On the length of Borel hierarchies](http://www.math.wisc.edu/~miller/res/hier.pdf) that it is consistent relative ZFC that no universal analytic set $U \subset [0,1] \times [0,1]$ belongs to the product $\sigma$-algebra $\mathcal{P}[0,1] \otimes \mathcal{P}[0,1]$. Combined with Rao's result mentioned by Michael Greinecker, this shows that $2^{\mathfrak{c \times c}} = 2^{\mathfrak{c}} \otimes 2^\mathfrak{c}$ is independent of ZFC.

See my answer to [Universally measurable sets of $\mathbb{R}^2$](http://math.stackexchange.com/q/177416) on math.stackexchange.com for related results and more details and references.