I am taking a probability theory course, and I need help understanding a homework question.
Let $(\Omega , \mathcal{F}, \mu)$ be a measure space, and $f:\Omega\rightarrow[0,\infty)$ be such that $\int{f\:\text{d}\mu} = 1$. Show that $v(E) := \int\mathbb{I}_E(\omega)f(\omega)\mu(\text{d}\omega), \: E \in \mathcal{F}$ defines a probability distribution on $(\Omega,\mathcal{F})$.
I went to speak to my professor, and he pointed me towards the definitions of a probability measure and probability space in my textbook. We also covered Lebesgue integration, but I do not know if this applies here. Also, there was some talk about converting these integrals to sums, but I'm quite lost. If anyone can point me in the right direction with explanations, or provide readings or references, I would be grateful.