If $F \subset R^d$ is a set of positive measure, and $F_x = \{y \in R^{d-1} | (x,y) \in F \}$, then by Fubini's theorem we know that $|F| = \int_{R} |F_x| dx$. If $X = \{ x \in R | F_x \neq \emptyset\}$ then is it true that for almost every $x \in X$, $|F_x| > 0$?
Cross-sectional measures of a set of positive measure
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real-analysis
measure-theory
lebesgue-integral
1 Answers
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Take $d=2$, $F= \{(x,x)\}_{x \in [0,1]} \cup [1,2]\times [0,1]$.
Then $mF = 1$ and $F_x = \begin{cases} \{x\}, & x \in [0,1] \\ [0,1], & x \in [1,2] \\ \emptyset, & \text{otherwise}\end{cases}$, and $X = [0,2]$.
However, $mF_x = 0$ for $x \in [0,1)$.