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

  • #1
Let E in R2 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.

------------------------------------

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 R2, 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 R2 whose horizontal section is a Borel subset of R is a sigma-algebra on R2 containing all the open sets. Which doesnt quite agree with my approach.

Any help would be appreciated.
 

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