What does $d\mathbb{P}(\omega)$ in integral mean? What does $d\mathbb{P}(\omega)$ under integral sign mean?
Like $$\int_B Xd\mathbb{\mathbb{P}}(\omega)$$
Can somebody explain? How can we integrate $X$ with respect to  $\mathbb{P}(\omega)$ where $\mathbb{P}$ is some prob. measure?
 A: The formula in your post is ill-formed. One can use one of the equivalent $$\int_B X(\omega)d\mathbb{\mathbb{P}}(\omega),\qquad\int_B Xd\mathbb{\mathbb{P}}.$$ One also finds the alternative $$\int_B X(\omega)\mathbb{\mathbb{P}}(d\omega).$$
A: A probability measure can be looked at as a function $\mathbb{P}:\mathcal{A}\rightarrow\left[0,1\right]$
where $\mathcal{A}$ is a $\sigma$-algebra. The function has some
nice properties like $\mathbb{P}\left(\bigcup_{n=1}^{\infty}A_{n}\right)=\sum_{n=1}^{\infty}\mathbb{P}\left(A_{n}\right)$
when the $A_{n}$ are disjoint elements of $\mathcal{A}$. There is
a one to one correspondence beween elements of $\mathcal{A}$ and
their characteristic functions $1_{A}$ so equally well we could look
at $\mathbb{P}$ as a function on characteristic functions. This opens
the door to expanding the domain of $\mathbb{P}$. We can make it a function on
so-called $\mathcal{A}$-measurable functions, that takes values in
$\mathbb{R}$. Again it has nice properties. If $X$ is such
a measurable function then $\mathbb{P}\left(X\right)$ could be practicized as a notation of its value under $\mathbb{P}$. However, the notation $\mathbb{P}\left(X\right)$
is (mostly) not practicized, and $\mathbb P$ is not looked at as a function on functions (why not?). Instead one uses notations like $\int Xd\mathbb{P}$
, $\int X\left(\omega\right)d\mathbb{P}\left(\omega\right)$ or $\int X\left(\omega\right)\mathbb{P}\left(d\omega\right)$, and speaks of integrals. Here $\int_{B}Xd\mathbb{P}$ is
a notation for $\int X.1_{B}d\mathbb{P}$.
