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A question about probability measure theory

  1. Jun 18, 2013 #1
    Hi all,
    I have a question about measure theory:
    Suppose we have probability space [tex](\mathbb{R}^d,\mathcal{B}^d,\mu)[/tex] where [tex]\mathcal{B}^d[/tex] is Borel sigma algebra.
    Suppose we have a function
    [tex]u:\mathbb{R}^d\times \Theta\rightarrow \mathbb{R}[/tex] where [tex] \Theta\subset\mathbb{R}^l,l<\infty[/tex] and [tex]u[/tex] is continuous on [tex]\mathbb{R}^d\times \Theta[/tex].
    Now consider the function [tex]G:\Theta\rightarrow[0,1][/tex] defined as follows:
    [tex]G(\theta)=\int\limits_{\epsilon\in\mathbb{R}^d}\mathbb{I}\{u(\epsilon, \theta)\geq 0\} \mu (d\epsilon)[/tex]
    where [tex]\mathbb{I}\{P\}[/tex] is an indicator function equal to 1 if P is true and 0 otherwise.
    Is [tex]G(\theta)[/tex] continuous on [tex]\Theta[/tex]?

    If you know the answer, could you please also tell me what kind of math books I need to look to find more about this? I would like to more about this by reading such text.

    Thanks a lot!
     
  2. jcsd
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