Measure Zero Sets: Proving \sigma(E) Has Measure Zero

[Kindayr]

Homework Statement

Let $\sigma (E)=\{(x,y):x-y\in E\}$ for any $E\subseteq\mathbb{R}$. If $E$ has measure zero, then $\sigma (E)$ has measure zero.

The Attempt at a Solution

I'm trying to show that if $\sigma (E)$ is not of measure zero, then there exists a point in $E$ such that $\sigma (\{e\})$ that has positive measure. But i don't know if this actually proves the question.

I have already shown that if $E$ open or a $G_{\delta}$ set, then $\sigma (E)$ is also measurable. Can I use these to solve this?

Any help is appreciated.

 

[Dick]

The set you've defined, $\sigma (E)$ is a subset of R^2. $\sigma (\{e\})$ has zero measure. It's a line in R^2.