Problem 1462 (difficulty: 8/10)

(a) Show that if \(\displaystyle A\subset\R^p\) is measurable and \(\displaystyle \lambda(A)>0\), then \(\displaystyle A-A\) contains a ball centered at the origin. (Steinhaus)

(b) Show that if \(\displaystyle A, B\subset\R^p\) are measurable with positive measure, then \(\displaystyle A+B\) has a nonempty interior.

(c) Show that if \(\displaystyle A\subset\R^p\) measurable with positive measure and \(\displaystyle B\subset\R^p\) has positive outer measure, then \(\displaystyle A+B\) has a nonempty interior.

Give me another random problem!