I learned recently that lines such as the boundary of some Julia sets and the Hilbert curve have area. I was wondering what the strict definition of such an area would be, and I intuitively came up with the following:
Let $S$ be the set of all points on the line.
Let $R$ be the set of points in some contiguous region containing the line fully inside of it.
The area of the line is the percent of $R$ also in $S$ multiplied by the area of the region.
Your definition suffers from the following(fatal, I would say) flaws:
You rely on the word “percent” to define your “area”. But you clearly don’t mean percent in the usual sense, because the word “percent” is (usually) defined for finite quantities and has something to do with amount out of $100$ if you like.
Your choice of the enclosing region R is not unique. Thus you must give a clear indication of how R is chosen and moreover prove that different choices of R give the same value of area.
The usual definition of area of S is $\mu(S) $where by $\mu$ I denoted Lebesgue measure(I guess you could use other measures as well). You can read more about it on wiki of course.