Are there uncountably many disjoint uncountable real null sets? It is easy to think of a countable family of disjoint Cantor sets, and their union is of course a null set. 
It is equally trivial to define an uncountable family of Cantor sets but, how can it be ensured that they are pairwise disjoint? 
Would their union be a null set?

The answer is yes: Let $C_r\subset \mathbb R^2$ for $r\in\mathbb R$ be defined as $$C_r=\{(r,c)\mid c\in \mathcal C\}$$(where $\mathcal C$ is the Cantor set). Then, consider a continuous surjection from $\mathbb R\to\mathbb R^2$. The inverse images of each $C_r$ are clearly uncountable and disjoint, and because the function is continuous, they are still zero sets.
