Let E in(adsbygoogle = window.adsbygoogle || []).push({}); R^{2}be a Borel Set. Show that all horizontal and vertical sections

{ x : (x, y) in E }, { y : (x, y) in E }

of E are Borel subsets ofR.

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I think I'm missing something out. My argument is that E is Borel, so E is formed of finitely many unions, intersections and complements of open sets inR^{2}, each of which has horizontal and vertical sections which are open inR(I have already shown this). Therefore the horizontal and vertical sections of E are formed of countably many unions, intersections and complements of open subsets ofR, thus they are Borel.

Am I wrong?

I have been advised to approach this with a view to showing that the family of subsets ofR^{2}whose horizontal section is a Borel subset ofRis a sigma-algebra onR^{2}containing all the open sets. Which doesnt quite agree with my approach.

Any help would be appreciated.

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# I need inspiration toshow that one-dimensional subsets of Borel sets in R^2 are Borel

Can you offer guidance or do you also need help?

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