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.