Set used as a "mask" to "clip" that set

Question

I am very interested to understand this phrase from wikipedia:

"Intuitively, this condition means that the set E must not have some curious properties which causes a discrepancy in the measure of another set when E is used as a "mask" to "clip" that set."

I understand that this is used to describe the Caratheodory condition: $\lambda^*(A)=\lambda^*(A\cap E)+\lambda^*(A\cap E^c)$.

How does the "mask" and "clip" come about?

Thanks!

Answer 1

"Clip" here is used in the sense of computer graphics -- to selectively draw visual elements that are not obstructed (or clipped) by an opaque foreground element. A similar use appears in signal processing -- where very large or very small values are replaced with pre-specified extreme values. In this usage, the range of the function is constrained to a set and points whose image would be outside the set are mapped to specified points in the set. These uses are based on the verb "to clip" meaning to cut away, as in gardening.

"Mask" is used in the same sense as described above for computer graphics. In this case, the foreground object masks the background object and (partially) prevents the background object being rendered. A similar use with a more physical basis is photomasking, where a physical obstruction is used to prevent certain wavelengths of light reaching a chemical reaction which would etch a semiconducting substrate -- the net result is that a negative image of the mask is etched from the semiconductor. A similar idea from digital image manipulation is "image masking", where "transparent"/"opaque" data from one array is used to select which members of another array are used in an image combining operation. This use is based on a long-time meaning of the word "mask": an object which obscures all or part of a face.

When you write "$A \cap E$", you are using the set $E$ to clip the set $A$ -- only those elements of $A$ that pass through the set $E$ (by being the same elements) appear in the result. When you write "$A \cap E^\mathrm{c}$", you are using the set $E$ to mask the set $A$ -- those elements in $A$ that are blocked by $E$ (by being the same elements) do not appear in the result.