I am reading Basic Stochastic Processes by Brzezniak and Zastawniak and I have come across an integral which does not make sense as a Riemann integral but I am unsure what type of integral it is then. It comes from the following definition
If there is a Borel function $f_\xi:\mathbb{R}\rightarrow\mathbb{R}$ such that for any Borel set $B\subset\mathbb{R}$ $$P(\{\omega\in\Omega: \xi(\omega)\in B\}) =\int_B f_\xi(x)dx$$ then $\xi$ is said to be a random variable with an absolutely continuous distribution and $f_\xi$ is called the density of $\xi$.
Since we are integrating over sets I would have thought it was a Lebesgue integral but in other books they always give the measure and don't just leave $dx$ which has me confused.
Yes, this is the Lebesgue integral restricted to the Borel $\sigma$-algebra. When dealing with functions on $\mathbb R^n$ it is fairly common for authors to denote integrals with respect to the Lebesgue measure (or restricted version) by using $dx$ instead of $d\lambda(x)$.