TITLE: Lebesgue Measure: show that a subset of R is equal to R
QUESTION [5 upvotes]: If B is a subset of R such that
I) B' has Lebesgue measure zero
II) B is closed under addition
Show that B = R
this is my first course in measure theory.  I only know that nonempty close and open subset of R is equal to R.  
What do I have from i and ii?  And what property of R should I use?

REPLY [2 votes]: For any real $x$, $B$ and $B - x = \{b - x : b \in B\}$ have full measure so there is some $y \in B \cap B - x$. Hence $y = z - x$ for some $z \in B$. Hence $y, z \in B$ and $x = z - y$. Hence $x \in B$.
This avoids Steinhaus theorem which shows that this is also true if $B$ had just positive measure.