Every countable union of rectangles in R2 is a Lebesgue measurable set. Is the converse true, too?


Specifically, I wonder whether the following statement is true:


Let A be a set in the unit square that is Lebesgue measurable. Then there a countable collection of rectangles and a null set such that A is equal to the union of the rectangles and the null set.



Characterization of Measureable Sets
No, see [here](http://en.wikipedia.org/wiki/Smith-Volterra-Cantor_set) for a counter-example (it's a variant of the Cantor set which has non null Lebesgue measure and does not contain any interval).


And if you want a counter-example for $\mathbb{R}^2$ instead of $\mathbb{R}$, just cross it with an interval.


