Characterization of Measureable Sets [closed]

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.

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closed as too localized by Bill Johnson, fedja, Qiaochu Yuan, Ryan Budney, Emil JeřábekDec 7 2011 at 11:19

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