# Show whether E is a Measurable Set

• October 13th 2011, 08:23 AM
Markeur
Show whether E is a Measurable Set
Let $(E_k)_{k=1}^{\infty}$ be a sequence of measurable sets in $\mathbb{R}^n$. Let $E$ be a subset of $\mathbb{R}^n$ so that $x \in E$ if and only if $x$ belongs to exactly 2 of the sets $E_k$. Determine whether E is measurable. If it is, show it. If not, give a counterexample.

I suppose it is measurable? No? Can't think of any counterexample.

• October 13th 2011, 08:30 AM
Georgii
Re: Show whether E is a Measurable Set
for any two positive integers $1\leq i consider the set $A_{i,j}=E_i\bigcap E_j\bigcap (\bigcap_{k\ne i,j}\mathbb{R}^n\setminus E_k)$. Such sets are measurable and $E$ is a union of such sets
• October 13th 2011, 08:51 AM
Hartlw
Re: Show whether E is a Measurable Set
Quote:

Originally Posted by Markeur
Let $(E_k)_{k=1}^{\infty}$ be a sequence of measurable sets in $\mathbb{R}^n$. Let $E$ be a subset of $\mathbb{R}^n$ so that $x \in E$ if and only if $x$ belongs to exactly 2 of the sets $E_k$. Determine whether E is measurable. If it is, show it. If not, give a counterexample.

I suppose it is measurable? No? Can't think of any counterexample.