Meaning of Point Evaluation

Question

I read in some general measure theory books and there is always like "define measure $x$ to be the point evaluation at $y$..." but when I look around online and some other books there is no mention on what is "point evaluation". Can anyone explain to me what is point evaluation?

Answer 1

"Point evaluation" is not a good name for a measure. The name of the measure is the Dirac measure. Point evaluation describes what happens when you integrate against it: namely, you get

$$\int_X f(x) \, d \mu = f(y)$$

(evaluation at the point $y$).