This maybe a non-sense question -- but just to make sure:
Suppose I prove something about some arbitrary set in $\mathbb{R}$ and find that the set is closed. Because I know the set is closed, it's automatically Borel and can be "assigned" a number using Lebesgue measure. This is correct, right?
I'm asking because I'm proving something about the image of a set under a function. I'm worried that the images can't be measured in the range. But I know that the image is closed.