# Thread: Show whether E is a Measurable Set

1. ## Show whether E is a Measurable Set

Let $\displaystyle (E_k)_{k=1}^{\infty}$ be a sequence of measurable sets in $\displaystyle \mathbb{R}^n$. Let $\displaystyle E$ be a subset of $\displaystyle \mathbb{R}^n$ so that $\displaystyle x \in E$ if and only if $\displaystyle x$ belongs to exactly 2 of the sets $\displaystyle E_k$. 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.

Thanks in advance.

2. ## Re: Show whether E is a Measurable Set

for any two positive integers $\displaystyle 1\leq i <j$ consider the set $\displaystyle A_{i,j}=E_i\bigcap E_j\bigcap (\bigcap_{k\ne i,j}\mathbb{R}^n\setminus E_k)$. Such sets are measurable and $\displaystyle E$ is a union of such sets

3. ## Re: Show whether E is a Measurable Set

Originally Posted by Markeur
Let $\displaystyle (E_k)_{k=1}^{\infty}$ be a sequence of measurable sets in $\displaystyle \mathbb{R}^n$. Let $\displaystyle E$ be a subset of $\displaystyle \mathbb{R}^n$ so that $\displaystyle x \in E$ if and only if $\displaystyle x$ belongs to exactly 2 of the sets $\displaystyle E_k$. 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.

Thanks in advance.
In my opinion any question about "something" should start off with a definition of "something" and a demonstration that the definition is satis fied. So here is my little contribution as a matter of principle from http://en.wikipedia.org/wiki/Measure_(mathematics)

σ-algebra, meaning that unions, intersections and complements of sequences of measurable subsets are measurable.

Let Σ be a σ-algebra over a set X. A function μ from Σ to the extended real number line is called a measure if it satisfies the following properties:
• Non-negativity:
for all

• Null empty set:

I would find it interesting to see how this would be done in the case of the previous post.