0
$\begingroup$

This question has come up while playing around with the Steinhaus theorem:

Let $F-F$ denote the algebraic difference $\{f-g \mod 1 | f,g \in F\}$. Suppose that $F\subset[0,1]$ with $\mu F>0$ , where $\mu$ is the usual Lebesgue measure. If we know that $F-F = [0,1]$, may we conclude that $\mu F = 1$?

Thanks for your input!

  • 0
    @Lubin, OP asks for example with positive Lebesgue measure. Of course, your example is easily modified.2012-11-12

1 Answers 1

1

Here is a simple counter-example: $F=[0,\frac{1}{2}]$.

  • 0
    Thanks richard! Should've seen something so straightforward.2012-11-12