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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?

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Could it mean that the measure of a set $A$ is 1 if $y\in A$, and 0 otherwise? – Hagen von Eitzen Sep 2 '12 at 18:45
up vote 4 down vote accepted

"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 $y$).

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In other words, we're thinking of the measure as corresponding to a linear functional on some space of functions, and so "point evaluation at $y$" refers to the linear functional $f \to f(y)$. – Robert Israel Sep 2 '12 at 19:25
@QiaochuYuan - Thanks for pointing out that this is a Dirac Measure. – Sandra Sep 2 '12 at 19:30
And very likely one only attempts to evaluate this functional on a space (e.g., continuous functions with sup norms on compacts) such that this functional is continuous. – paul garrett Sep 2 '12 at 19:40

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