Let $\{X_r\}_{r\in[0,1]}$ be i.i.d. random variables, each distributed uniformly on $[0,1]$. Let $S\subseteq[0,1]^2$ be the random set defined as follows:
$$S=\{(r,X_r)\mid r\in[0,1]\}$$
How would such a set look like? Will it be "space filling"? Formally, will it have Hausdorff dimension 2 (with high probability)? And if so, will it have (with high probability) a 2-dimensional Hausdorff measure 1?