- #1
sephiseraph
- 6
- 0
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.
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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 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 doesn't quite agree with my approach.
Any help would be appreciated.
{ 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 doesn't quite agree with my approach.
Any help would be appreciated.