You can't measure something without borders because what you measure is the distance between the borders. Infinity cannot have a size because size deals with finite objects. If infinity had a size there could be a bigger size, and then it wouldn't be infinity because something would be bigger than it.
Let's assume that infinity can be measured, and has been. Let's call it x. What about x+1, x*2, and x^99999999? Infinity cannot have a size. A size exists within something larger. If infinity had a size, that would mean it exists within something larger.
Infinity is not a thing.