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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!

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# A question about probability measure theory

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