If a set has positive measure, do its slices have positive measure?

Question

Suppose $F \subset R^n$ and $|F|>0$. Let $F_x = \{ y \in R^{n-1} | (x, y) \in F\}$. Then is it true that for almost every $(x,y) \in F$, $F_x$ has positive $n-1$ dimensional measure?

By Fubini's Theorem, we know that $|F| = \int_{R} |F_x| dx$, but I can't see why this implies the truth of the stated fact. All I can gather from this is that $|F_x|$ is positive on a subset of $R$ which has positive linear measure.

Answer 1

Think it from the other side: consider the set $A\subset F$ of points $(x,y)\in F$ such that the section $F_x$ has $(n-1)$-measure zero. Then by Fubini $A$ has $n$-measure zero.

(Some sections could be non-measurable, but a.e. section will be, again by Fubini).