space of finite borel measure is the dual of continuous function vanishing at infinity I have a question. Why the space of finite borel measure is the dual of the space of continuous fucntions that vanish at infinity? 
If we have a finite borel measure, then any continuous function vanishing at infinity integrate with respect to this measure will get a finite number. So the space of borel measure should be a subset of the dual. But why all of it? 
Here is what I think: So given a linear functional on the space of continuous fucntion vanishing at infinity, in partically, the linear functional can act on the mollified indicator funtion of disjoint bounded open cubes. Then we can assign every such open set a number, then constructing a borel measure. But it seems this linear fucntional can take value infinity. I got confused. Maybe this is not the right way to do so. Thanks for any help!

The idea you mention in your last paragraph is precisely the idea behind the proof of the Riesz Representation Theorem (Riesz-Markov, or Riesz-Markov-Kakutani depending on the source). 
What makes your objection disappear is the boundedness of the functional. For details, you will have to check the proof of the theorem, which is not short. 
