We know and accept that a line in a 2-d space has zero width. However, this causes a an issue as explained here.
Suppose that we have a closed area $A$ of White color, and we draw one colored line (of any other color other than White) crossing it. The consumed area by the line from the area of the shape is zero and the remaining White area (without the are consumed by the line) of the shape is still $A$. Now if we draw a very large number of colored lines crossing the White closed area, we know that the total free area $A$ will be affected. This can be simply simulated by pencil and paper. We can say that:
White Area = $A$ - (Number of Lines) * (Area of each colored Line)
But since the "Area of each colored line" is zero, this means that the White area value will never be affected regardless how many lines cross it. However this is contrary to what we can experiment with a paper and a pencil.
How can this be represented mathematically and justified?
A sample with few lines is shown below.