Showing the sum of two sets contains an interval - Baire's Theorem? 
If $E$ and $F$ are measurable subsets of $\mathbb{R}$ and $m(E), m(F) >0,$ then $E+F$ contains an interval. 

The path to the standard solution to this is built on the notion that a measurable set of positive measure contains "almost a whole interval" (Stein and Shakarchi, "Real Analysis," p. 44). 
However, having recently studied the Baire Category Theorem for the first time, I wonder if there's any way to prove this using that famous result. My thought was maybe to assume the theorem is false, recognize any countable union of closed sets that equals $E+F$ implies each set in the union does not contain an open interval, and then focus on some kind of particular union that would force one of $E$ or $F$ to not be measurable, i.e. a Vitali set of some type. I'm not sure how to proceed - or whether that line of thought is even valid, since as far as I can tell Baire's Theorem has little reference to measurability. 
I apologize for the over-generality of the question, but I think it would help me understand Baire's Theorem and its proper setting a bit more. 
 A: I am now convinced it is impossible to use BC arguments to prove this (without measure theory).  First of all, you write "assume the theorem is false..." and then try to use Baire category assuming $E+F$ does not contain an interval.  
This is a misunderstanding; just because a set does not contain any interval does not say anything about its Baire category: it may be 1, it may be 2.  A set $D$ can be dense, uncountable, Baire category 2, intersecting every set in full measure (i.e. $m(D\cap S)=m(S)$ for every measurable $S$) and still not contain any interval.  Example: the irrational numbers.  Thus unfortunately your proof sketch cannot work.
To see the mechanics of why BC does not see the problem of sumsets, consider two sets $E$ and $F$ of BC 2 and positive measure; you can always find $E'\subset E$ and $F'\subset F$ of positive measure and BC 1 (exercise), so already an interval $I$ exists with $I\subset E'+F'\subset E+F$, so the fact that $E$ and $F$ are BC 2 is not relevant for the existence of an interval in the sumset.
The above in particular implies that the addition map $E\times F\to E+F$ does not preserve category, so information about BC cannot be communicated between the source and the target.
