How to show this function is measurable?

How do I show that $$\int_{\Omega(t)} |\sum_{i=1}^n f_i(t)g_i(t)|^p$$ is measurable wrt $t \in [0,T]$, where $f_i(t)$ is continuous wrt $t \in [0,T]$ and $g_i(t)=\mathbf{1}_{B_i}(t)$ is measurable wrt $t \in [0,T]$ (the $B_i$ are dissjoint and measurable)?

Also $f_i(t) \in \Omega(t)$ for all $t$.

Note that domain of integration depends on $t$. Make any assumptions necessary.

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Are $f,g$ also functions of $\omega\in\Omega(t)$? What's supposed to be under the absolute value, and what role does the sum play? – Jonathan Y. Aug 20 '13 at 10:05
@Yes they're functions of $\omega.$ Sorry I edited the abs value. I guess the sum doesn't matter. I just icnluded it as that's what i have. – aere Aug 20 '13 at 10:10
But then surely there are $f_n,g_n$ as well? Also, I'm still not clear on the function's domain, if you don't mind fleshing it out a little. – Jonathan Y. Aug 20 '13 at 10:14
Yes you're right. I edited. Note the $g_n$ are constant wrt. $\Omega(t)$. Thanks for your attention. – aere Aug 20 '13 at 10:27
I'm sorry, I should've been clearer. I'm asking regarding the integration variable. Do we have, for all $n,t$, that $f_n(t):\Omega(t)\to\mathbb{C}$ is continuous (in $\omega$, given some natural topology, say), or something else? How would you formulate what you're getting at? – Jonathan Y. Aug 20 '13 at 10:37