I want to find some subset of R^2 which its intersection with every vertical line is measure zero if we see it as a subset of R and it is not measure zero in R^2?
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It is a well known result of Sierpinski that there exists nonmeasurable subsets of $\mathbb{R}^{2}$ which intersect each line in at most two points. Furthermore, there exists a realvalued function whose graph is a nonmeasurable subset of $\mathbb{R}^{2}$.


