Let $P\subset \mathbb{R}^2$ be a positive Lebesgue measure set. Then $P$ may not contain a subset of the form $A\times B$ where $A,B\subset \mathbb{R}$ are of positive Lebesgue measure.

For example consider $P=\{(x,y)\in [0,1]\times[0,1]:x-y\notin \mathbb{Q}\}.$

This example leads me to ask:

Given any $P\subset \mathbb{R}^2,$ a positive Lebesgue measure set, does there exists a measure zero set $U\subset \mathbb{R}^2$ such that $P\cup U$ contains a subset of the form $A\times B$ where $A,B\subset \mathbb{R}$ are of positive Lebesgue measure?

Let $P\subset \mathbb{R}^2$ be a positive Lebesgue measure set. Then $P$ does not necessarily contain a subset of the form $A\times B$ where $A,B\subset \mathbb{R}$ are of positive Lebesgue measure.

For example consider $P=\{(x,y)\in [0,1]\times[0,1]:x-y\notin \mathbb{Q}\}.$

This example leads me to ask:

Given any $P\subset \mathbb{R}^2,$ a positive Lebesgue measure set, does there exists a measure zero set $U\subset \mathbb{R}^2$ such that $P\cup U$ contains a subset of the form $A\times B$ where $A,B\subset \mathbb{R}$ are of positive Lebesgue measure?