I would like to ask a question about the product (Lebesgue) measure on rectangle. I tried to solve the problem but I couldn't.

Let $S$ be a subset of a region, say $R$ which is enclosed by a rectangle. Assume that if any line, say $L$ parallel to each side (both horizontal and vertical lines) of the rectangle, then $L \cap S$ has full (Lebesgue) measure on $L$.

Does the set $S$ have the full measure on $R$ with respect to the product (Lebesgue) measure? If yes, how could I show it?

Thank you for your answer or any book or references.

I would like to ask a question about the product (Lebesgue) measure on rectangle. I tried to solve the problem but I couldn't.

Let $S$ be a subset of a region, say $R$ which is enclosed by a rectangle. Assume that if any line, say $L$ parallel to each side (both horizontal and vertical lines) of the rectangle, then $L \cap S$ has full (Lebesgue) measure on $L$.

1. If $S$ is measurable, does $S$ have the full measure on $R$?
2. If $S$ is non-measurable, does $S$ have to contain a (proper) subset which has the full measure on $R$ with respect to the product (Lebesgue) measure? If yes, how could I show it?

Thank you for your answer or any book or references.