Suppose $(\Omega, \mathcal{F}, \mu)$ is a probability space and suppose I have a random variable $X: \Omega \rightarrow \mathbb{R}$. By definition, we have $$\mathbb{E}[X]=\int_{\Omega}Xd\mu=\int_{\Omega}X(\omega)\mu(d\omega)$$
This question might sound stupid but I feel confused about notation:
Is it wrong to use $d\mu(\omega)$ instead of $\mu(d\omega)$ and in general, is there any difference?
In general, if I have a random variable $X$, is it true that for any measurable function $f$, one has $$\mathbb{E}[f(X)]=\int f(x)d\mu(x)$$
In particular, can I write the expression above as
$$\mathbb{E}[X]=\int_{\Omega}xd\mu(x)$$
I would intuitively say yes, but even though I am ashamed of myself, I can't write down a formal proof
Hints, tips and suggestions are all welcome