Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space. Let $X$ be random variable on that space mapping to $(\mathcal{X}, \mathcal{C})$. So when integrating with respect to the distribution $\mathbb{P}_X$ of the RV $X$, for instance when computing the expectation of $X$, we get a sum if $X$ is a discrete RV.
1.) I have always worked with this, however, I actually do not know how to prove this. E.g., how to show the last equality of $$\int_{\Omega} X(\omega) \mathbb{P}(d\omega)= \int_{\mathcal{X}}x\mathbb{P}_X(dx)=\sum_{x\in\mathcal{X}}x\mathbb{P}_X(x). $$
2.) If $X$ is again as defiend above but not necessarily a discrete RV and we want to integrate over a set that is a singleton (and f is a measurable function), e.g. $$\int_{\{a\}}f(x)\mathbb{P}_X(dx)$$ How to actually compute this? Is there a difference if $X$ is discrete or if $X$ is absolutely continuous?