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**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 of

**R**.

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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 in

**R**

^{2}, each of which has horizontal and vertical sections which are open in

**R**(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 of

**R**, thus they are Borel.

Am I wrong?

I have been advised to approach this with a view to showing that the family of subsets of

**R**

^{2}whose horizontal section is a Borel subset of

**R**is a sigma-algebra on

**R**

^{2}containing all the open sets. Which doesnt quite agree with my approach.

Any help would be appreciated.