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?
Meaning of Point Evaluation
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measure-theory
definition
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3Could it mean that the measure of a set $A$ is 1 if $y\in A$, and 0 otherwise? – 2012-09-02
1 Answers
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"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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2In 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 $f \to f(y)$. – 2012-09-02
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0@QiaochuYuan - Thanks for pointing out that this is a Dirac Measure. – 2012-09-02
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0And 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_. – 2012-09-02