A line of slope $a$ is placed in the plane for every real number $a$ (these lines need not pass through the origin.) Does the set of points not on any of these lines have finite area?
I'm not sure if the phrasing in this problem is precisely correct, so please correct me if my meaning is unclear.
Obviously the area is not always $0$; for example, choosing all lines tangent to a given circle leaves the interior of the circle untouched.
The lines can leave a zero, finite or infinite area uncovered. (I suspect they can also leave an unmeasurable set uncovered.)
If the lines go through the origin, the entire plane is covered.
If the lines are tangent to a semicircle, an infinite strip on the other side of the circle is left uncovered.
It's slightly less straightforward to leave a finite area uncovered, but it's possible. The lines
$$ \pmatrix{t\\0}+\lambda\pmatrix{\sin\frac t{1+t}\alpha\\\cos\frac t{1+t}\alpha}\;, $$
where $\lambda$ is the line parameter and $t$ parametrizes the family of lines, only use the angle range $[0,\alpha]$, cover the entire first and second quadrants but leave the entire third quadrant uncovered. Three such families starting out at the edges of a triangle can be made to have non-overlapping angle ranges and to cover everything except for the triangle; the remaining slopes can then be “wasted” e.g. by making them tangent to the triangle's circumcircle.