Restriction of a null set

Question

Suppose you have a null set, $S$ in $\mathbb{R}^n$. Is it true than in that case, there always exists an immersion $i: \mathbb{R} \hookrightarrow \mathbb{R}^n $ such that for almost all $x \in \mathbb{R} $, $i(x)$ does not intersect $S$ (so $i(\mathbb{R}) \cap S$ is a null set in $i(\mathbb{R})$)?

Answer 1

By Fubini, for almost all hyperplanes $H_c:= \{\overline{x}: x_n=c\}$, we have $S\cap H_c$ null in $H_c$. Then apply the same argument to one such $H_c$, repeating until the hyperplanes are just lines. The result is uncountably many straight lines that are the image of the immersion you want.