The notation
<br />
\int_\Omega X(\omega}) \, \mathcal{P}(dw)<br />
is used in probability to indicate the expectation of the random variable X
with respect tot the probability measure (distribution) \mathcal{P} over the probability space \Omega.
If \Lambda is any measurable set, then
<br />
\int_\Lambda X(\omega) \, \mathcal{P}(dw) = E[X \cdot 1_{\Lambda}]<br />
If the probability space is the real line with measure \mu, then
<br />
\int_\Lambda X(\omega) \, \mathcal{P}(dw) = \int_\Lambda f(x) \, \mu(dx)<br />
is the Lebesgue-Stieltjes integral of f with respect to the
probability measure \mu.
In more traditional form, if F is the distribution function of \mu, and \Lambda is an interval (a,b), then
<br />
\int_\Lambda X(\omega) \, \mathcal{P}(dw) = \int_\Lambda f(x) \, \mu(dx) = \int_{(a,b)} f(x) \, dF(x)<br />
If the probability measure doesn't have any atoms, the final integral is just a Lebesgue integral. If there are atoms, you need to take care to specify the interval according to whether the endpoints are or are not included - e.g.
<br />
\int_{a+0}^{b+0} f(x) \,dF(x), \quad \int_{a-0}^{b-0} f(x) \, dF(x)<br />
and so on.
Billingsley is one of the "classic" probability texts. Chang's "A Course in Probability Theory" is another - I studied from it many years ago, and have the second edition. His writing is a little terse, but there is a lot packed into his book.