A question about probability measure theory

[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!

 

[mmwave]

Hi there,

Thank you for your question. To answer your first question, yes, G(\theta) is continuous on \Theta. This can be proven using the continuity of u on \mathbb{R}^d\times \Theta and the continuity of \mu on \mathcal{B}^d. Specifically, for any \theta_0\in\Theta and any \epsilon>0, we can find a \delta>0 such that for all \theta\in\Theta, if |\theta-\theta_0|<\delta, then |u(\epsilon,\theta)-u(\epsilon,\theta_0)|<\epsilon. This means that for any \theta_0, if we take a small enough neighborhood around \theta_0, the values of u(\epsilon,\theta) will be close to u(\epsilon,\theta_0) for all \epsilon\in\mathbb{R}^d. And since \mu is continuous, the integral will also be close to the value at \theta_0. Therefore, G(\theta) is continuous on \Theta.

To learn more about this topic, I recommend looking into measure theory textbooks such as "Probability and Measure" by Patrick Billingsley or "Real Analysis" by Royden and Fitzpatrick. These texts cover the basics of measure theory and can provide a deeper understanding of concepts such as probability spaces, Borel sigma algebras, and integrals. Additionally, you may also find resources on mathematical analysis and functional analysis helpful in understanding the continuity of functions. I hope this helps and good luck with your studies!