# A question about probability measure theory

1. Jun 18, 2013

### hwangii

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

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!