# Does a set with strictly positive Lebesgue measure contain an interval?

I am studying a function whose Fourier transform is zero on a set of strictly positive Lebesgue measure and I need to know this:

If a set has a strictly positive Lebesgue measure can we prove that it contains an interval?

Help is much appreciated

• How about the irrationals? Jun 10 '16 at 21:43

• @JosephDoob No, because a fat Cantor set always undershoots the measure of the interval in which it is contained by some $\epsilon$.